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Early  American  Mathematics  Books 


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ELEMENTS  OF  ALGEBRA: 


ON 


THE  BASIS  OF  M.  BOURDON: 


EMBRACINO 


STURM'S    AND    HORNER'S    THEOREMS, 


PRACTICAL  EXAMPLES, 


BY    OHAELES    DAYIES,   LL.D. 

AUTHOR     OF     ARITHMETIC,    ELEMENTARY    ALGEBRA,   ELEMENTARY    GEOMETRY,   PRACTICAL 

GEOMETRY,    ELEMENTS   OF  SURVEYING,   ELEMENTS  OF  DESCRIPTIVE  AND 

ANALYTICAL   GEOMETRY,   ELEMENTS  OF  DIFFERENTIAL 

AND   INTEGRAL  CALCULUS,    AND   A  TREATISE 

ON    SHADES,    SHADOWS  AND    PER- 

8PE0TIVB. 


NEW    YOEK: 
A.  S.  BARNES  &  CO.,  Ill  &  113   WILLIAM  STREET, 

(corner  OP  JOHN  STREET.) 

80LI>   BY  BOOKSELLERS,  GENERALLY,  TIIROUGnOUT  THE  UNITED  STATES. 

1868. 


Babies' 


©abfes*  primacy  ^nt1)metic  and  STable^^^Soofe— Designed  for  Beginners; 
containing  the  elementary  tables  of  Addition,  Subtraction,  Multiplication, 
Division,  and  Denominate  Numbers  ;  with  a  large  number  of  easy  and  prac- 
tical questions,  both  mental  and  written. 

f9a\)ies*  j^ivBt  Wessons  in  ^ritljmettc— Combining  the  Oral  Method  with  the 
Method  of  Teaching  the  Combinations  of  Figures  by  Sight. 

JBabics'  intellectual  ^vitljmetic— An  Analysis  of  the  Science  of  Numbers,  with 
especial  reference  to  Mental  Training  and  Development. 

Uabies*  Xeto  Scljool  ^ritbmetic— Analytical  and  Practical. 

3S.C3)  to  23 allies*  T^eto  Scljool  ^vit^metic. 

©abies*  Grammar  of  ^ritf)metic— An  Analysis  of  the  Language  of  Numbers 
and  the  Science  of  Figures. 

Babies*  Neb)  Slnibersit^  ^rit|)metic— Embracing  the  Science  of  Numbers,  and 
their  Applications  according  to  the  most  Improved  Methods  of  Analysis  and 
Cancellation. 

l^ej)  to  ©abies*  'Ne\ii  S^nibersit^  0[vit|)metic. 

23abies'  Hlcmentar^  Algebra— Embracing  the  First  Principles  of  the  Science. 

2te5  to  Babies*  fSlementar^  ^llsebra. 

30abies*  SElementar^  CKecmetv^  and  STrisonometrg— With  Applications  in 
Mensuration. 

Babies*  practical  Platl)ematifS— With  Drawing  and  Mensuration  applied  to 
the  Mechanic  Arts. 

JBabfes*  S^tuibersitn  0lQebta— Embracing  a  Logical  Development  of  the 
Science,  with  graded  examples. 

©abies*  3Souvtion*s  Algebra— Including  Sturm's  and  Horner's  Theorems, 
and  practical  examples. 

Babies*  3!lrgcn'Dre*s  ^eometrg  antJ  STrfgonometrs—Pevised  and  adapted  to 
the  course  of  Mathematical  Instruction  in  the  United  States. 

Babies*  JBUmenlB  of  Surbejjing  and  "Mabigatlon — Containing  descriptions 
of  the  Instruments  and  necessary  Tables. 

Babies*  ^nnlgtical  CKeometvi>— Embracing  the  Equations  of  the  Point,  the 
Straight  Line,  the  Conic  Sections,  and  Surfaces  of  the  first  and  second  order. 

Babies*  Bifferential  and  integral  Calculus* 

Babies*  Bescrijptibe  (^Keometr^— With  its  application  to  Spherical  Trigonome- 
try, Spherical  Projections,  and  Warped  Surfaces. 

Babien*  Sbaties,  Sl)atrob3s,  and  ^.Sers^ectibe. 

BabiexJ*  3logic  antr  TOih'ti)  of  l^atbematics—With  the  best  methods  of  In- 
struction Explained  and  Illustrated. 

Babies*  anti  J^tck*s  l^atbematical  Bictfonarj)  anti  @:i)clo})etifa  of  f^atte*- 
mat'cal  Science — Comprising  Definitions  of  all  the  terms  employed  m 
Mathematics— an  Analysis  of  each  Branch,  and  of  the  whole,  as  forming  a 
Ringle  Science. 

Entered  according  to  Act  of  Congress,  in  the  year  one  thousand  eight  hundred  and  fifty- 
one,  by  Charles  Davies,  in  the  Clerk's  Oflice  of  the  District  Court  of  the  United  State* 
for  the  Southern  District  of  New  York. 

"William  Denysk,  STEUEOTYPaB  and  Eleotrotypkr,  183  William  Street,  New  York. 


WUm 


PREFACE 


The  Treatise  on  Algebra,  by  M.  Bourdon,  is  a  work 
of  singular  excellence  and  merit.  In  France,  it  has 
long  been  one  of  the  standard  Text  books.  Shortly  after 
its  first  publication,  it  passed  through  several  editions, 
and  has  formed  the  basis  of  every  subsequent  work  on 
the  subject  of  Algebra,  both  in  Europe  and  in  this  country. 

The  original  work  is,  however,  a  full  and  complete 
treatise  on  the  subject  of  Algebra,  the  later  editions 
containing  about  eight  hundred  pages  octavo.  The  time 
which  is  given  to  the  study  of  Algebra,  in  this  country, 
even  in  those  seminaries  where  the  course  of  mathe- 
matics is  the  fullest,  is  too  short  to  accomplish  so  volu- 
minous a  work,  and  hence  it  has  been  found  necessary 
either  to  modify  it  essentially,  or  to  abandon  it  alto- 
gether. 

In  the  following  work,  the  original  Treatise  of  Bourdon 
has  been  regarded  only  as  a  model.  The  order  of  ar- 
rangement, in  many  parts,  has  been  changed;  new  rules 
and  new  methods  have  been  introduced:  the  modifica- 
tions indicated  by  its  use,  for  twenty  years,  as  a  text  book 


4:  PKEFACE. 

in  the  Military  Academy  have  been  freely  made,  for 
the  purpose  of  giving  to  the  work  a  more  practical 
character,  and  bringing  it  into  closer  harmony  with  the 
trains  of  thought  and  improved  systems  of  instruction 
which  prevail  in   that  institution. 

But  the  work,  in  its  present  form,  is  greatly  indebted 
to  the  labors  of  William  G.  Peck,  A.  M.,  TJ.  S-  Topo- 
graphical Engineers,  and  Assistant  Professor  of  Mathe- 
matics in  the  Military  Academy. 

Many  of  the  new  definitions,  new  rules  and  improved 
xTiiethods  of  illustration,  are  his.  His  experience  as  a 
teacher  of  mathematics  has  enabled  him  to  bestow  upon 
the  work  much  valuable  labor  which  will  be  found  to 
bear  the  mark*  of  profound  study  and  the  freshness  of 
daily  instruction. 

FiSEKILL     LANDrUQ,  i 

May,  1868.       f 


CONTENTS. 


CHAPTER  I. 

DEFINITIONS    AND    PRELIMINARY   REMARKS. 

Algebra— Definitions — Explanation  of  the  Algebraic  Signs 1 — 28 

Similar  Terms — Reduction  of  Similar  Terms c 28—80 

Theorems — Problems — Definition  of — ^Problem 30 — 3 1 

CHAPTER  II. 

ADDITION,    SUBTRACTION,  'MULTIPLICATION,    AND    DIVISION. 

Addition— Rule 81—3* 

Subtraction — Rule — Remark 35 — 4-1 

Multiplication — Rule  for  Monomials  and  Signs 41 — 15 

Rule  for  Polynomials 45 — 46 

Remarks — Theorems   Proved 46 — 49 

Division  of   Monomials — Rule 49 — 53 

Signification  of  the   Symbol  a^ 53 — 55 

Division  of  Polynomials — Rule *. 55 — 58 

Remarks  on  the  division  of  Polynomials 58 — 59 

Of  Factoring   Pylynomials 59 — 60 

When  m  is  entire,  «"» — b"^  is  divisible  by  a — 6... •  60 — 62 

CHAPTER  III. 

ALGEBRAIC    FRACTIONS. 

Definition — ^Entire  Quantity — ^Mixed  Quantity , . .  62 — 68 

Reduction  of  Fractions c 68 — 69 

To  Reduce  a  Fraction  to  its   Simplest  Form 68 — L 

To   Reduce  a  Mixed  Quantity  to  a  Fraction 68 — II. 

To   Reduce  a  Fraction  to  an  entire  or  Mixed  Quantity 68 — TIL 

To   Reduce  Fractions  to  a  Common   Denominator 68 — IV 

To  Add   Fractions 68— V. 

To  Subtract  Fractions , 68— VI 


6  CONTENTS. 

ARTlCLBa 

To  Multiply  Fractiors 68— VIL 

To  Divide   Fractions 68— VIII 

Results  from  adding  to  both   Terms   of  a   Fraction 10 — 71 

Syitbols  0,  00  and; U— ^2 

CHAPTER  IV. 

EQUATIONS    OF    THE    FIRST    DEGREE. 

Definition  of  an  Equation — Different  Kinds — Properties  of  Equations  12 — 77 

Solution  of  Equations , 77 — 78 

Tiansformation  of  Equations — First   and   Second 78 — 80 

Resolution  of   Equations  of  the   First   Degree — Rule 81 

Problems   involving  Equations  of  the   First  Degree 81 

Equations   with   two   or  more    Unknown   Quantities 82 — 83 

Elimination — By  Addition — By  Subtraction — By  Comparison 83 — 88- 

Problems   giving  rise   to   Simultaneous   Equations...,^ Page  96 

Indeterminate   Equations   and  Indeterminate   Problems 88 — 89 

Interpretation  of  Negative   Results ; 89 — 91 

Discussion  of  Problems 91 — 92 

Inequalities 92—93 

CHAPTER  V. 

EXTRACTION    OF    THE  SQUARE  ROOT  OF  NUMBERS. OF  ALGEBRAIC  QUAN- 
TITIES.-.—TRANSFORMATION    OF    RADICALS  OF    THE    SECOND    DEGREE. 

Extraction  of  the   Square   Root  of  Numbers 93 — 96 

Extraction  of  the  Square  Root  of  Fractions 96 — 100 

Extraction  of  the   Square   Root  of  Algebraic  Quantities 100 — 104 

Of  Monomials 100—101 

Of  Polynomials 101—104 

Radicals  of  the   Second  Decree 104 — 106 

Addition  and  Subtraction — Of  Radicals 106 — 107 

Multiplication,  Division,  and  Transformation ...  107 — 110 

CHAPTER  VI. 

EQUATIONS    OF    THE    SECOND    DEGREE. 

Equations  of  the   Second  Degree 110 — 112 

Incomplete  Equations — Solution  of 112 — 1 14 

Solution  of  Complete   Equations  of  the  Second  Degree 114 — 1 16 

Discussion  of  Equations  of  the   Second  Degree 115 — 117 

Of  the  Four  Forms 117—121 

Problem  of  the   Lights 121—122 

Of  Trinomial  Equations 122—126 

Extraction  of  the  Square   Root  of  the   Binomial  a  ±y/ b 125 — 126 

Equations  with  two  or  more   Unknown  Quantities r  . . . .   • .  * .  3  26 — 128 


CONTENTS.  7 

CHAPTER  VII. 

FORMATION     OP     POWERS,    BINOMIAL     THEOREM,    EXTRACTION     OF    ROOTS 
OF    ANY    DEGREE    WHATEVER. OF   [RADICALS. 

Formation  of  Powers,,  • « , 128 — 13G 

Olieory  of  Permutations   and   Combinations 130 — 136 

Binomial  Theorem 136—141 

Extraction  of  the   Cube  Roots  of  Numbers 141 — 142 

To  Extract  the  w'*  Root  of  a   Whole  Number. , 142—144 

Extraction  of  Roots  by  Approximation 144 — 145 

Extraction  of  the  n*^  root  of  Fractions 145 — 146 

Cube   Root  of  Decimal  Fractions. 146— 14t 

Extraction  of  Roots  of  Algebraic   Quantities 147 — 14S 

Of  Polynomials 148—150 

Transformation  of  Radicals 150 — 15^ 

Addition  and   Subtraction  of  Radical*. 155—166 

Multiplication  of  Radicals ,    .- 156 — 151 

Division  of  Radicals 157 — 158 

Formation  of  Powers  of  Radicals ,    158 — 159 

Extraction  of  Roots 159 — 160 

Different   Roots  of    the   Same  Power 1 60 — 162 

Modifications  of  the   Rules  for   Radicals 162 — 164 

Theory  of  Fractional   and  Negative   Exponents .164 — 111 

CHAPTER  VIII. 

OF    SERIES. ARITHMETICAL    PROGRESSION. GEOMETRICAL    PROFORTION 

AND      PROGRESSION. RECURRING      SERIES. BINOMIAL      FORMULA. 

SUMMATION    OF    SERIES. PILING    OF    SHOT    AND    SHELLS. 

Series  Defined V\\ — 172 

Arithmetical   Progression — Defined 172 — 173 

Expression  for  the   General   Term 174 — 176 

Sum  of  any  two  Terms 175 — 176 

Sum  of  all  the  Terms 176—177. 

Formulas   and   Examples 177 — 181 

Ratio   and   Geometrical   Proportion 181 — 186. 

Geometrical   Progression — Defined 186 — ]  87 

Expression  for   any   Term , 187 — ]  S8 

Sum  of  n  Terms — Formulas   and   Examples 188 — 193 

Indeterminate  Co-efficients 1 93 — 199 

Recurring  Series , ,    199 — 202 

General   Demonstration  of  Binomial  Theorem. , 202 — 204 

Applications  of  the   Binomial  Formula , 204 — 208 

Summation  of  Series , 208 — 209 

Method  of  Diffejgnces 209—210 

Piling  of  Balls ....,     216-215 


8  CONTENTS. 

CHAPTER  IX. 

CONTINUED     FRACTIONS. EXPONENTIAL     QUANTITIES. LOGARITHMS.— 

Continued   Fractions 215 — 224 

Exponential  Quantities 224 — 22*7 

Theory  of  Logarithms 227—229 

General   Propertites  of  Logarithms , 229 — 236 

Logarithmic  Series — Modulus , 236 — 241 

Ti'ansformation  of  Series 241 — 242 

Of  Interpolation 242—243 

Uf  Interest < 243—244 

CHAPTER  X. 

GENERAL  THEORY  OF  EQUATIONS. 

General   Properties  of  Equations , 244 — 251 

Composition  of  Equations 261 — 252 

Of  the   Greatest   common   Divisor 262 — 262 

Transformation  of  Equations 262 — 264 

Formation  of  Derived  Polynomials 264 — 266 

Properties  of   Derived   Polynomials 266 — 267 

Equal   Roots 267—270 

Elimination 270—276 

CHAPTER  XI. 

SOLUTION    OF    NUMERICAL    EQUATIONS. STURm's     THEOREM. CARDAn's 

RULE. HORNEr's      METHOD. 

General  Principles 275 — 277 

First   Principle 277 — 279 

Second   Principle 279—280 

Third  Principle 280— 28 1 

Limits  of  Real  Roots 281—284 

Ordinary  Limits  of  Positive  Roots 284 — 285 

Smallest  Limit  in   Entire   Numbers 285 — 286 

Superior  Limit  of  Negative   Roots — Inferior  Limit  of  Positive  and 

Negative   Roots 286—287 

Consequences 287 — 293 

Descartes'  Rule 293 — 296 

Commensurable   Roots  of  Numerical  Equations. ,,, 295 — 298 

Sturm's  Theorem 298—308 

Cardan's  Rule ..,..........,, 308—309 

Preliminaries   to   Horner's   Method 309 — 3 1 0 

Multiplication  by   Detached  Co-efficients 810 — 311 

Division  by  Detached  Co-efficients » .311 — 312 

Synthetical  Division ^ 312—313 

Method  of  Transformation 313—314 

Homer's  Method .,  c    314 


INTRODUCTION. 


Quantity  is  a  general  term  applicable  to  everytKing  which 
can  be  increased  or  diminished,  and  measured.  There  are  two 
kinds  of  quantity;, 

1st.  Abstract  quantity,  or  quantity,  the  conception  of  which 
does  not    involve    the   idea  of    matter ;   and, 

2dly.  Concrete  quantity,  wliich  embraces  every  thing  that  is 
material. 

Mathematics  is  the  science  of  quantity ;  that  is,  the  science 
which  treats  of  the  measurement  of  quantities,  and  of  their 
relations  to   each  other.      It  is   divided  into   two   parts : 

1st.  The  Pure  Mathematics,  embracing  the  principles  of  the 
science  and  all  explanations  of  the  processes  by  which  these 
principles  are  derived  from  the  abstract  quantities.  Number 
and    Space :    and, 

2d.  The  Mixed  Mathematics,  embracing  the  applications  of 
these  principles  to  all  investigations  involving  the  laws  of 
matter,  to  the  discussion  of  all  questions  of  a  practical  nature, 
and  to  the  solution  of  all  problems,  whether  they  relate  to 
abstract  or   concrete   quantity.* 

*Davies'  Logic  and  Utility  of  Mathematics.    Book  XL 


10  INTRODUCTION. 

• 

There  are  three  operations  of  the  mhid  which  are  iiT.rne 
diately  concerned  in  the  investigations  of  mathematical  science : 
Isti.  Apprehension;    2d.  Judgment;    3d.  Reasoning. 

1st.  Apprehension  is  the  notion,  or  conception  of  an  idea 
in   the    mind,  analogous   to   the   perception   by  the    senses. 

2d.  Judgment  is  ,  the  comparing  together,  in  the  mind,  two 
of  the  ideas  which  are  the  objects  of  Apprehension,  and  pro 
uounciiig  that  they  agree  or  disagree  with  each  other.  Judg 
inent,  therefore,  is   either   affirmative  or  negative. 

3d.  Reasoning  is  the  act  of  proceeding  from  one  judgment 
lo  another,  or  of  deducing  unknown  truths  from  principles  al- 
ready known.  Language  affords  the  signs  by  which  these  opeia- 
tions  of  the  mind  are  expressed  and  communicated.  An  appre 
hension,  expressed  in  language,  is  called  a  term;  a  judgment, 
expressed  in  language,  is  called  a  proposition;  and  a  pro/. ess 
of  reasoning,  expressed  in  language,  is  called  a  demonsira- 
tion* 

The  reasoning  processes,  in  Logic,  are  conducted  usually  by 
means  of  words,  and  in  all  complicated  cases,  can  take  place 
in  no  other  way.  The  words  employed  are  sians  of  \deas^ 
and  are  also  one  of  the  principal  instruments  or  helps  of 
thought;  and  any  imperfection  in  the  instrument,  or  in  the 
mode  of  using  it,  will  destroy  all  ground  of  confidence  in  the 
result.  So,  in  the  science  of  mathematics,  the  meaning  of  the 
terms  employed  are  accurately  defined,  while  the  language 
arising  from  the  use  of  the  symbols,  in  each  branch,  has  a- 
ilefinite   and   precise   signification. 


*  Whatelj's   Logic, — of  tte   operations  of  the  mind  and   senses. 


INTRODUCTION.  11 

In  the  science  of  numbers,  the  ten  characters,  called  figures, 
are  the  alphabet  of  the  arithmetical  language ;  the  combinations 
of  these  characters  constitute  the  pure  language  of  arithmetic; 
and  the  principles  of  numbers  which  are  unfolded  by  means 
of  this,  m  connection  with  our  common  language,  constitute 
the  science. 

In  Geometry,  the  signs  which  are  employed  to  indicate  the 
boundaries  and  forms  of  portions  of  space,  are  simply  the 
straight  line  and  the  curve;  and  these,  in  connection  with  our 
common  language,  make  up  the  language  of  Geometry :  a 
science  which  treats  of  space,  by  comparing  portions  of  it 
with  each  other,  for  the  purpose  of  pointing  out  their  proper 
ties   and   mutual   relations. 

Analysis  is  a  general  term  embracing  that  entire  portion  of 
mathematical  science  in  which  the  quantities  considered  are 
represented  by  letters  of  the  alphabet,  and  tho^  operations  to 
be  performed   on   them  are   indicated   by  signs. 

Algebra,  which  is  a  branch  of  Analysis,  is  also  a  species 
of  universal  arithmetic,  in  which  letters  and  signs  are  employed 
to  abridge  and  generalize  all  processes  involving  numbers.  It 
is  divided  into  two  parts,  corresponding  to  the  science  and 
art  of  Arithmetic : 

1st.    That   which    has   for   its   object   the   investigation  of    the 

,  properties  of  numbers,  embracing  all  the-  processes  of  reasoning, 

by  which  new   properties   are   inferred  from   known   ones ;    and, 

2d.    The   solution  of  all  problems   or   questions   involving  the 

determination    of    certain    numbers    which    are    unknown,    from 

their   connection   with   certain  others  which   are  known  or  given. 


12  INTKODUCTION. 

In  arithmetfc,  all  quantity  is  regarded  as  coEJsisting  of  parts, 
which  can  be  numbered  exactly  or  approximatively,  and  in 
this  respect,  possesses  all  the  properties  of  numbers.  Proposi- 
tions, therefore,  concerning  numbers,  have  this  remarkable  pecu 
liarity,  that  they  are  propositions  concerning  all  quantities 
whatever.  Algebra  extends  the  generalization  still  further.  A 
number  is  a  collection  of  things  of  the  same  kind,  without  refer- 
ence to  the  nature  of  the  thing,  and  is  generally  expressed  by 
figures.  Algebraic  symbols  may  stand  for  all  numbers,  or  for  all 
quantities  which  numbers  represent,  or  even  for  quantities  which 
cannot  be  exactly  expressed  numerically. 

In  Geometry,  each  geometrical  figure  stands  for  a  class ; 
and  when  we  have  demonstrated  a  property  of  a  figure,  that 
property  is  considered  proved  for  every  figure  of  the  class.  In 
Algebra,  all  numbers,  all  lines,  all  surfaces,  all  solids,  may  be 
denoted  by  a  single  symbol,  a  or  x.  Hence,  the  conclusions 
deduced  by  means  of  those  symbols  are  true  of  all  things  what- 
ever, and  not  like  those  of  number  and  Geometry,  true  only 
for  particular  classes  of  things.  The  symbols  of  Algebra,  there- 
fore, should  not  excite  in  our  minds  ideas  of  particular  things. 
The  written  characters,  a,  5,  c,  d,  x^  y,  ^,  serve  as  the 
representatives  of  things  in  general,  whether  abstract  or  con- 
crete, whether  known    or   unknown,  whether   finite    or   infinite. 

In  the  various  uses  which  we  make  of  these  symbols,  aid 
the  processes  of  reasoning  carried  on  by  means  of  them,  the 
mind  insensibly  comes  to  regard  them  as  things^  and  not  as 
mere  signs ;  and  we  constantly  predicate  of  them  the  properties 
of  things   in    general,  without  pausing   to   inquire   what   kind  of 


INTRODUCTION.  18 

thing  is  implied.  All  this  we  are  at  liberty  to  do,  since  the 
symbols  being  the  representatives  of  quantity  in  general,  there 
is  no  necessity  of  keeping  the  idea  of  quantity  continually  alive 
in  the  mind;  and  the  processes  of  thought  may,  without  dan- 
ger, be  allowed  to  rest  on  the  symbols  themselves,  and  there- 
fore, become  to  that  extent,  merely  mechanical.  But  when  we 
look  back  and  see  on  what  the  reasoning  is  based,  and  how 
the  processes  have  been  conducted,  we  shall  find  that  every 
step  was  taken  on  the  supposition  that  we  were  actually 
dealing  with  things,  and  not  with  symbols;  and  that  without 
this  understanding  of  the  language,  the. whole  system  is  without 
signification,  and  fails.* 

The  quantities  which  are  the  subjects  of  the  algebraic  analysis 
m^y  be  divided  into  two  classes :  those  which  are  known  or 
given,  and  those  which  are  unknown  or  sought.  The  known 
are  uniformly  represented  by  the  first  letters  of  the  alphabet, 
a,  6,  c,  c?,  &c. ;  and  the  unknown  by  the  final  letters,  x,  y, 
2,  V,   &c. 

Five  operations,   only,    can  be    performed    upcii    a    quantity 
that   will   give  results   difiering  from   the  quantity  itself:    viz. 
1st.  To   add   a   quantity  to   it; 
2d.  To  subtract  a  quantity  from  it; 
3d.  To  multiply  it  by  a  quantity; 
4th.  To  divide  it ; 
5th.  Tc  extract  a  root  of  it. 

Five  signs  only,  are  employed  to  denote  these  operations. 
They  are   too  well    known   to   be   repeated   here.      These,    with 

•  Davies*  Logic  and  Utility  of  Mathematics,    g  278. 


14  INTRODUCTION. 

the  signs  of  equality  and  inequalitj,  together  with  the  letters  of 

fhe  alphabet,  are    the    elements  of  the   algebraic   language. 

The    interpretation    of    the   language    of    Algebra    is   the   first 

ihing   to    which   the   attention  of    a    pupil    should   be   directed; 

and  he   should   be   drilled    in   the   meaning   and    import  of  the 

symbols,  until    their    significations  and   uses   are   as   familiar   as 

the    sounds  of  the   letters  of  the  alphabet. 

« 
All   the   apprehensions,  or   elementary  ideas,  are    conveyed    to 

the    mind   by   means    of    definitions    and    arbitrary    signs  ;     and 

every  judgment  is  the  result  of  a  comparison  of  such  impressions. 

Hence,  the  connection  between  the  symbols  and  the  ideas  which 

vhey  stand   for,  should   be    so    close    and   intimate,  that   the    one 

^hall    always    suggest    the    other;     and    thus,    the    processes    of 

Algebra  become  chains  of  thought,  in  which  each  link  lulfils  the 

double  ofitco  of  a  distinct  and  comiecting  propo£  tioQ. 


ELEMENTS  OF  ALGEBRA. 


CHAPTER  I. 

DEFINITIONS    AND    PRELIMINARY-    REMARKS. 

1.  Quantity  is  anything  which  can  be  increased  or  dimifi- 
khed,  and   measured. 

2a  Mathematics  is  the  science  which  treats  of  the  measurement 
and  relations  of  quantities. 

3.  Algebra  is  a  branch  of  mathematics,  in  which  the  quantities 
considered  are  represented  by  letters,  and  the  operations  to  be 
performed  upon  them  are  indicated  by  signs.  The  letters  azid 
signs  are  called  symbols. 

4.  in  algebra  two  kinds  of  quantities  are  considered: 

1st.  Known  quantities^  or  those  whose  values  are  known  or 
given.  These  are  represented  by  the  leading  letters  of  the  alplia- 
bet,  as,  a,  5,  c,  &c. 

2d,  Unknown  quantities^  or  tjiose  whose  values  are  not  given. 
They  are  denoted  by  the  final  letters  of  the  alphabet,  as, 
a;,  y,  2,  &;c. 

Letters  employed  to  represent  quantities  are  sometimes  written 
with  one  or  more  dashes,  as,  a\  h'\  c"\  x\  y"^  &c.,  and  are 
read,  a  prime,  b  second,  c  third,  x  prime,  y  second^  &;c. 

5.  The  sign  4-,  is  called  plus,  and  when  placed  between  two 
quantities,  indicates  that  the  one  on  the  right  is  to  be  added  to 
the  03ie  on  the  left.     Thus,  a  +  6  is  read  a  p]us  h,  and  indicates 


16  ELEMENTS   OF  ALGEBRA^  [CHAP.   I. 

tliat  the  quantity  represented  by  h  is  to  be  added  to  the  quan- 
tity represented  by  a. 

6.  The  sign  — ,  is  called  minus,  and  when  placed  between  two 
quantities,  indicates  that  the  one  on  the  right  is  to  be  subtracted 
from  the  one  on  the  left.  Thus,  c  —  d  is  read  c  minus  c?,  and 
indicates  that  the  quantity  represented  by  d  is  to  be  subtracted 
fi'om  the  quantity  represented  by  c. 

The  sign  +,  is  sometimes  called  the  positive  sign,  and  the 
quantity  before  which  it  is  placed  is  said  to  be  positive. 

The  sign  —,  is  called  the  negative  sign,  and  quantities  affected 
by  it  are  said  to  be  negative. 

7.  The  sign  X ,  is  called  the  sign  of  multiplication,  and  when 
placed  between  two  quantities,  indicates  that  the  one  on  the  left 
is  to  be  multiplied  by  the  one  on  the  right.  Thus,  a  x  b,  indi 
eates  that  a  is  to  be  multiplied  by  b.  The  multiplication  of 
quantities  may  also  be  indicated  by  placing  a  simple  point 
between  them,  as  a.b,  which  is  read  a  multiplied  by  b. 

The    multipli'cation   of   quantities,   which    are    represented    by 
letters,  is  generally  indicated   by  simply  writing  the  letters  one 
after  another,  without  interposing  any  sign.     Thus, 
ob     is  the  same  as  a  X  b,  or  a.b; 
and  abc,    the  same  as  a  X  b  X  c,  or  a.b.c. 

It  is  plain  that  the  notation  last  explained  cannot  be  employed 
when  the  quantities  are  represented  by  figures.  For,  if  it  were 
required  to  indicate  that  5  was  to  be  multiplied  by  6,  we 
could  not  write  5  6,  without  confounding  the  product  with  the 
number  56. 

The  result  of  a  multiplication  is  called  the  product,  and  each 
of  the  quantities  employed,  is  called  a  factor.  In  the  product 
of  several  letters,  each  single  letter  is  called  a  literal  factor. 
Thus,  in  the  product  ab  there  are  two  literal  factors  a  and  b ;  in 
the  product  bed  there  are  three,  b,  c  and  d. 

8.  The  sign  -r,  is  called  the  sign  of  division,  and  when  placed 
between  two  quantities,  indicates  that  the  one  on  the  left  is  to  be 
divided  by  the  one  on  the  right.    Thus,  a  —  6  indicates  that  a  is  to 


CHAP.   I.]  DEFINITIONS  AND  REMARKS.  17 

be  divided  by  h.     The  same  operation  may  be  indicated  by  writing 

a 
b  under  a,  and  drawing  a  line  between  them,  as  —  ;  or  by  writing 

h  on  tho  right  of  a,  and  drawing  a  line  between  them,  as  a\h, 

9.  The  sign  =,  is  called  the  sign  of  equality^  and  indicates  that 
tne  two  quantities  between  which  it  is  placed  are  equal  to  each 
other.  Thus,  a  —  h  ^^  c  -\-  d^  indicates  that  a  diminished  by  6  is 
equal  to  c  increased  by  d. 

10.  The  sign  >,  is  called  the  sign  of  inequality^  and  is  used  to 
indicate  that  one  quantity  is  greater  or  less  than  another. 

Thus,  a  >  6  is  read,  a  greater  than  h  ;  and  a  <  6  is  read,  a  less 
than  h ;  that  is,  the  opening  of  the  sign  is  turned  toward  the  greater 
quantity. 

11.  The  sign  '^  is  sometimes  employed  tc  indicate  the  difference 
>f  two  quantities  when  it  is  not  known  which  is  the  greater. 

Thus,  a  /^  5,  indicates  the  difference  between  a  and  6,  without 
showing  which  is  to  be  subtracted  from  the  other. 

12.  The  sign  oc,   is  used  to  indicate  that,  one  quantity  varies  as 

to  another.     Thus  a  oc    -r-,  indicates  that  a  varies  as  -7-. 
0  0 

13.  The  signs  :  and  :  :,  are  called  the  signs  of  proportion;  tliB 
first  is  read,  is  to,  and  the  second  is  read,  as.     Thus, 

a  :  b  :  :  c  :  d^ 
is  read,  a  is  to  5,  as  c  is  to  d.  ^ 

The  sign   .*.,   is  read  hence^  or  consequently, 

14«  If  a  quantity  is  taken  several  times,  as 
a-\-a-\-a-\'a-\-a^ 
it  is   generally  written  but  once,   and  a  number  is  then  placed 
before  it,  to  show  how  many  times  it  is  taken.     Thus, 

a-\-  a  +  a  -\-  a  -r  a   may  be  written    5a. 
The  number  5  is  called  the  co-efficient  of  a,  and  denotes  tliat  a  is 
taken  5  times. 

Hence,  a  co-efficient  is  a  number  prefixed  to  a  quantity  denoting 
the  number  of  times  which  the  quantity  is  taken. 

2 


18  ELEMENTS   OF  ALGEBRA.  [CHAP.   L 

When  no  co-efficient  is  written,  the  co -efficient  1  is  always  under- 
stood;  thus,    a    is  the  same  as    la.  • 

15.  If  a  quantity  is  taken  several,  times  as  a  factor,  the  product' 
may  be  expressed  by  writing   the   quantity  once,  and  placing   a 
number  to  the  right  and  above  it,  to  show  how  many  times  it  18 
taken  as  a  factor. 

Thus,  axaXaXaXa   may  be  written   aK 

The  number  5   is  called   an   exponent^  and  indicates   that  a  is 

taken  5  times  as  a  factor. 

Hence,  an  exponent  is  a  number  written  to  the  right  and  above 

a  quantity,  to  show  how  many  times  it  is  taken  as  a  factor.     If 

no  exponent  is  written,  the  exponent  1  is  understood.     Thus,  a  is 

the  same  as   o}, 

16.  If  a  quantity  be  taken  any  number  of  times  as  a  factor,  the 
resulting  product  is  called  a  power  of  that  quantity :  the  exponent 
denotes  the  degree  of  the  power.     For  example, 

a^  =z  a  is  the  first  power  of  a, 
o?  z=:z  a  X  a  is  the  second  power,  or  square  of  a, 
a^  =za  X  a  X  a  is  the  third  power,  or  cube  of  a, 
a*=:aXaXaXa  is  the  fourth  power  of  a, 
a^  —  axaxaxaxa\s>  the  fifth  power  of  a, 
m  which  the  exponents  of  the  powers  are,  1,  2,  3,  4  and  5 ;  and 
the  powers  themselves,  are  the  results  of  the  multiplications.     It 
should  be  observed  that  \h.Q,  exponent  of  a  power  \^  always  greater 
by  one  than  the  number  of  multiplications.     The  exponent  of  a 
power  of  a  quantity  is  sometimes,  for  the  sake  of  brevity,  called 
the  exponent  of  the  quantity. 

17.  As  an  example  of  the  use  of  the  exponent  in  algebra,  let 
it  be  required  to  express  that  a  number  a  is  to  be  multiplied 
tliree  times  by  itself;  that  this  product  is  then  to  be  multiplied 
three  times  by  5,  and  this  new  product  twice  by  c  ;  we  should 
write 

axaxaxaxhxhxhxcxc=i  d^b^c^. 
If  it  were  further  required  to  take  this  result  a  ceriiain  numbef 
of  times,  say  seven,  we  should  simply  write  la'^Pi^ 


CHAP.   L]  DEFINITIONS  AND   REMAEKS.  19 

18#  A  root  of  a  quantity,  is  a  quantity  which  being  taken  a 
certain  number  of  times,  as  a  f^.ctor,  will  produce  the  given 
quantity. 

The  sign  .^/^is  called  the  radical  sign,  and  when  placed  over 
a  quantity,  indicates  that  its  root  is  to  be  extracted.     Thus, 
^y~a    or  simply  ^^/a    denotes  the  square  loot  of  a. 
£/a   denotes  the  cube  root  of  a. 
^Ta    denotes  the  fourth  root  of  a. 
The  number  placed  over   the   radical  sign  is  called   the   indpyx 
of  the  root.     Thus,  2  is  the  index  of  the  square  root,  3  of  tlio 
cube  root,  4  of  the  fourth  root,  &c. 

19t  The  reciprocal  of  a  quantity,  is  1  divided  by  that  quantity. 
Thus, 

—    is  the  reciprocal  of  a: 
a 

and  — — y    is  the  reciprocal  of  a  +  5. 

,.1  a-\-h 

20 •  Every  quantity  written  in  algebraic  language,  that  is,  by 
the  aid  of  Tetters  and  signs,  is  called  an  algebraic  quantity^  or  the 
algebraic  expression  of  a  quantity.     Thus, 

is  the  algebraic  expression  of  three  times  the 
quantity  denoted  by  a ; 
j     is  the  algebraic  expression  of  five  times  the 
t        square  of  a  ; 

j     is  the  algebraic  expression  of  seven  times  the 

(       product  of  the  cube  of  a  and  the  square  of  6; 

„        _ ,  j     is  the  algebraic   expression  of  the   difference 

(       between  three  times  a  and  five  times  h\ 

is  the  algebraic  expression  of  twice  the  square 

of  a,  diminished  by  three  times  the  'produet 

of  a  and   5,  •  augmented   by   four  times   the 

square  of  h, 

21.  A  single  algebraic  expression,  not  connected  with  any  other 
by  the  sign  of  addition  or  subtraction,  is  called  a  monomial^  oi 
«imply,  a  term. 


3a  I 


2«2 -306  +  452^ 


20  ELEMENTS   OF  ALGEBRA.     ^  [CHAP,    t 

Thus,      3a,     5a2,     Ta^^^,    are  monomials,  or  single  terms. 

An  algebraic  expression  composed  of  two  or  more  terms  cod* 
aected  by  the  sign  +  or  — ,  is  called  a  polynomial. 

For  example,    3a  —  55  and  2o?  —  3c5  +  45^,  are  polynomials. 

A  polynomial  of  two  terms,  is  called  a  binomial;  and  one  of 
three  terms,  a  trinomial, 

22.  The  numerical  value  of  an  algebraic  expression,  is  the  num 
ber  obtained  by  giving  a  particular  value  to  each  letter  which 
enters  it,  and  performing  the  operations  indicated.  This  numer- 
ical value  will  depend  on  the  particular  values  attributed  to  the 
letters,  and  will  generally  vary  with  them. 

For  example,  the  numerical  value  of  2a^,  will  be  54  if  we  make 
a  =  3;  for,     3^  =.:  3  X  3  X  3  =  27,  and  2  X  27  =  54. 

The  numerical  value  of  the  same  expression  is  250  when  we 
make  a  =  5  •  for,     5^  =r  5  X  5  X  5  ==  125,  and  2  X  125  =  250. 

We  say  that  the  numerical  value  of  an  algebraic  expression 
generally  varies  with  the  values  of  the  letters  which  enter  it;  it 
does  not,  however,  always  do  so.  Thus,  in  the  expre^ion  a  -^  b, 
so  long  as  a  and  b  are  increased  or  diminished  by  the  same 
number,  the  value  of  the  expression  will  not  be  changed. 

For  example,  make  a  =  7  and  5  =  4:  there  results  a  —  5  =  3. 

Now,  make  a  r=  7  -f-  5  =  12,  and  5  =  4  +  5  =  9,  and  there 
results,  as  before,  a  —  5  =  12  —  9  =  3. 

23 •  Of  the  different  terms  which  compose  a  polynomial,  some 
are  preceded  by  th6  sign  +,  and  others  by  the  sign  — .  The 
former  are  cftlled  additive  terms,  the  latter,  subtractive  terms. 

When  the  first  term  of  a  polynomial  is  plus,  the  sign  is  gene- 
rally omitted ;  and  when  no  sign  is  written  before  a  term^  it  is 
always  understood  to  have  the  sign  +. 

24.  The  numerical  value  of  a  polynomial  is  not  affected  by 
changing  the  order  of  its  terms,  provided  the  signs  of  all  the 
terms  remain  unchanged.     For  example,  the  polynomial 

4a'^  —  3a25  +  5ac2  =  5ac2  —  3^25  +  4a3  =  —  3a25  +  5ac2  _|.  4^^^ 

25.  Each  literal  factor  vrhich  enters  a  term,  is  called  a  dimen- 
sion of  the  term ;  and  the  degree  of  a  term  is  indici^ted  by  the 
number  of  these  factors  or  dimensions.     Thu«? 


CHAP.   I.]  DEFINITIONS  AND   KEMARKS.  2l 

3a    is  a  term  of  one  dimension,  or  of  the  first  degree. 

bab    is  a  term  of  two  dimensions,  or  of  the  second  degree. 

la%c^  =  laaahcc  is  of  six  dimensions,  or  of  the  sixth  degree. 

In  general,  the  degree  of  a  term  is  determined  by  taking  the  sum 
of  the  exponents  of  the  letters  which  enter  it.  For  example,  the 
term  Sa^bcd^  is  of  the  seventh  degree,  since  the  sum  of  the  expo- 
nents, 

2+1  +  1+3,    is  equal  to  7. 

26«  A  polynomial  is  said  to  be  homogeneous,  when  all  of  its 
terms  are  of  the  same  degree.     The  polynomial 

Sa  —  2b  +  c  is  homogeneous  and  of  the  first  degree. 

—  4a5  +  b^  is  homogeneous  and  of  the  second  degree. 

5d^c  —  4c3  +  2c'^d    is  homogeneous  and  of  the  third  degree. 

Sa^  —  4a5  +  c  is  not  homogeneous. 

27»  A  vinculum  ,  parenthesis   (),   brackets    [],    {  },  oi* 

bar  I,  may  be  used  to  indicate  that  all  the  quantities  which  they 
connect  are  to  be  considered  together.     Thus, 


a-i-  b  +  c  X  on,  {a  +  b  -\-  c)  X  X,  [a  +  b  +  cjxx,  or  {a-\-b  +  c}x, 
indicate  that  the  trinomial  a  +  6  +  c  is  to  be  multiplied  by  x. 

When  the  parenthesis  or  brackets  are  used,  the  sign  of  mul- 
tiplication may  be  omitted :  as,  (a  -\-  b  +  c)  x.  The  bar  is  used 
in  some  cases,  and  differs  from  the  vinculum  in  being  placed 
vertically,  as  +  a  x, 

+  c 

28.  Terms  which  contain  the  same  letters   affected  with  equal 

exponents  are  said  to  be  similar.     Thus,  in  the  polynomial, 

lab  +  Sab  -  4aW  +  ba%\ 

the  tenns  lab  and  Sab,  are  similar,  and  so   also   are  the  terms 

—  4:a^P  and  Sa^S^,  the  letters   in   each  being   the  same,  and  thft 

same   letters   being    affected   with   equal   exponents.     But  in  the 

binomial 

Sa^  +  lab\ 

the  terms  are  not  similar;  for,  although  they  contain  the  same 
letters,  yet  the  same  letters  are  not  affected  with  equal  expo- 
nents. 


22  ELEMENTS   OF  ALGEBRA.    •  [CHAP.   I, 

29.  When  a  polynomial  0)ntains  similar  ,erms,  it  may  be 
reduced  to  a  simpler  form  by  forming  a  single  term  from  each 
set  of  similar  terms.  It  is  said  to  be  in  its  simplest  form^  when 
it  contains  the  fewest  terms  to  which  it  can  be  reduced. 

If  we  take  the  polynomial 

we   know,   from    the    definition  of  a   co-efficient,   that   the  literal 
part  a^bc^  is   to   be  taken  additively,  2+6  +  11,  or    19    timfts; 
and  subtractively,  4  +  8,  or  12  times. 
Hence,  the  given  polynomial  reduces  to 

It  may  happen  that  the  corefficient  of  the  subtractive  term,  ob- 
tained  as  above,  will  exceed  that  of  the  additive  term.     In  that 
case,  subtract   the  positive   co-efficient  from  the  negative,  prefix  the 
minus  sign  to  the  remainder,  and  then  annex  the  literal  part. 
In  the  polynomial 

Za^b  +  2a%  -  5a^  -  Sa% 
we   have,  +  Sa^  —  5a^ 

+  2a^  -  Sa^ 


+  5a26  -  Sa^b 

But,  —  Sa^  =  —  5a2&  —  Sa^  :    hence 

5a26  -  Sa^  =:  5a25  -  5a^  -  Za^b  =z  -  Sa^J. 

In  like  manner  we  may  reduce  the  similar  terms  of  any  poly- 
nomial.  Hence,  for  the  reduction  of  a  polynomial  containing 
sets  of  similar  terms,  to  its  simplest  form,  we  have  the  following 

RULE. 

I.  Add  together  the  co-efficients  of  all  the  additive  terms  of  each  setf 
and  annex  to  their  sum  the  literal  part :  form  a  single  subtractive 
term  in  the  mme  manner. 

II.  Then,  subtract  the  less  co-efficient  from  the  greater,  and  to  the 
remainder  prefix  the  sign  of  the  greater  co-efficient,  aiid  annex  iht 
literal  part* 


CHAP.   I.]  REDUCTION   OF  POLYNOMIALS.  28 

EXAMPLES. 

1.  Reduce  the  polynomial  4a^b  —  Sa^b  —  9a^b  +  lia^b  to  \U 
simplest  form.  Ans.  —  2a^b. 

2.  Reduce  the  polynomial  labc^  —  abc^  —  Ifabc'^  —  Sabc^  +  (jabc* 
to  its  simplest  form.  Ans,   —  Zdbc^, 

3.  Reduce    the    polynomial     9cb^  —  Sac^  +  \bcb^  +  8ca  +  9ao^ 

—  24ci^    to  its  simplest  form.  Ans.  ac^  +  8ca. 

4.  Reduce  the  polynomial  Qac^  —  6ab^  +  lac^  —  SaS^  —  ISac^ 
4-  18a63    to  its  simplest  form.  Ans.  lOab^. 

6.  Reduce  the  polynomial  abc^  —-  abc  +  6ac^  —  9abc^  +  6ahc 
--  Sac^    to  its  simplest  form.  Ans.   —  Sabc^  +  5a6c  —  Sac^. 

6.  Reduce  the  polyfiomial  3^262  ^  ^a^i,  +  5^5  _  9a2^2  _|_  9^3^^ 
+  3aZ>   to   its   simplest  form.  Ans.   —  Qa^b^  +  2a36  +  Sab. 

7.  Reduce  the  polynomial  3ac5*  —  la^c^b^  —  Qa'^b^  —  Sa^b* 
H-  Ga^c^js  —  Qacb"^  +  4a46^  +  2a*Z>^   to   its   simplest   form. 

Ans.   —  a^c^^s  —  GacJ'^. 

8.  Reduce  the  polynomial   —  7aH^c^  +  da^bc^  +  ^aWc^  +  aWc^ 

—  5a*5c2  —  55^   to   its   simplest  form.  Ans.  Aa^bc^  —  b^o°. 

9.  Reduce  the  polynomial  —  \Qa%  +  Qa%'^  -^  la%  —  ba%'^ 
-  ba%  +  3^262  i^Q  its  simplest  form.         Ans.   —  Sa^  +  4a262. 

Remark. — It  should  be  observed  that  the  reduction  affects  only 
the  co-efficients,  and  not  the  exponents. 

30.  A  THEOBicM  is  a  general  truth,  which  is  made  evident  by  a 
course  of  re-asoning  called  a  demonstration. 

A  PROBLEM  is  a  question  proposed  which  requires  a  solution. 

31.  We  shall  now  illustrate  the  utility  and  brevity  of  algebraic 
lai^guage  by  solving  the  following 

PROBLEM. 

The  sum  of  two  numbers  is  67,  and  their  difference  is  19  ;  what 
are  the  numbers  ? 

Let  us  first  indicate,  by  the  aid  of  algebraic  symbols,  the 
relation  whicli  exists  between  the  given  and  unknown  numbers 
of  the  problem. 


24  ELEMENTS   OP  ALGEBRA.  [CHAP.   L 

If  the  less  of  the  two  numbers  were  k?iown,  the  greater  could 
be  found  by  adding  to  it  the  difference  19  ^    or  in  other  words, 
the  less  number,  plus  19,  is  equal  to  the  greater. 
If,  then,  we  denote  the  less  number  by  rr, 

a;  +  19   will  denote  the  greater, 
and  2^+19   will  denote  the  sum. 

But  from  the  enunciation,  this  sum  is  to  be  equal  to  '67.     Ther© 

2a;  -M9  =  67. 

Now,  if  2x  augmented  by  19,  is  equal  to  67,  2x  alone  is  equal 

t><)  67  minus  19,  or 

2x  =  67-  19, 

or  performing  the  subtraction,  , 

2x  =  48. 

Hence,    x  is  equal  to  half  of  48,  that  is 

48      ^, 
.==-  =  24. 

The  less  number  being  24,  the  greater  is 
a;  +  19  =  24  +  19  =:  43. 
And,  indeed,  we  have 

43  +  24  =  67,   and  43  -  24  =  19. 

GENERAL    SOLUTION. 

The  sum  of  two  numbers  is   a,  and  their  difference  is   h      What 
are  the  two  numbers  ? 

Let  X    denote  the  less  number  ; 

Then  will  x  +  b    denote  the  greater  number. 

Now,  from  the  conditions  of  the  problem, 
x-\'  X  +  b,    or    2x'{-b 
^ill  be   equal  to   the   sum   of  the  two  numbers  :    her?  je, 

2x  +  b  =  a. 
Now,  if  2x  +  b    is   equal  to  a,   2x   alone   must  be   equal   W 
H  "    h    and 

_^CL  —  b  __  a         b 


x  +  b  = 

a 
'"2  "" 

X 

^h. 

=i+ 

h 
2  ~ 

a 

b 

X 

"^2 

2  ~ 

CHAP.   I.J  SOLUTION  OF  PROBLEMS.  25 

If  the  value  of  x  be  increased  by  5,  we  shall  have  the 
greater  number :    that  is, 

-2+^  =  Y+2"' 
hemce,       ir  +  6  =  —  +  -^=    the  greater  number,  and 

■=z    the  less   number. 

That  is,  the  greater  of  two  numbers  is  equal  to  half  their  sum 
increased  by  half  their  difference  ;  and  the  less  is  equal  to  half 
their  sum  diminished  by  half  their  difference. 

As  the  form  of  these  results  is  independent  of  any  particular 
values  attributed  to  the  letters  a  and  6,  the  expressions  are  called 
formulas^  and  may  be  regarded  as  comprehending  the  solution 
of  all  problems  of  the  same  kind,  differing  only  in  the  numerical 
values  of  the  given   quantities.     Hence, 

A  formula  is  the  algebraic  expression  of  a  general  rule,  or 
principle. 

To  apply  these  formulas  to  the  case  in  which  the  sum  is  237 

sad  difference  99,  we  have 

237    .  99      237  +  99      336       ,    * 
the  greater  number  =  -^  ^'  "9"  = 9 —  "9"  —  -^"^  >     ^ 

^    ^    ,                       237       99      237-99      138 
and  the  less  =  -^^ —  = =  -^  :z=    69  ; 

and  these  are  the  true  numbers;   for, 

168  +  69  =i  237    which  is  the  given  sum, 
and  168  —  69  =    99    whicfc  is  the  given  differences 


CHAPTER  n. 

ADDITION,   SUBTI.  ACTION,   MULTIPLICATION,  AND  DIVISION. 


ADDITION. 

31  •  Addition,  in  algebra,  is  the  operation  of  finding  the  sim- 
plest equivalent  expression  for  the  aggregate  of  two  or  more  alge- 
braic quantities.     Such  equivalent  expression  is  called  their  sum, 

32 •  If  the  quantities  to  be  added  are  dissimilar,  no  reductions 
can  be  made  among  the  terms.  We  then  write  them  one 
after  the  other,  each  with  its  proper  sign,  and  the  resulting 
polynomial  will  be   the   simplest   expression   for   the   sum. 

For  example,  let  it  be  required  to  add  together  the  mono- 
mials . 

3a,    hh   and   2c ; 
we   connect  them   by   the   sign   of  addition, 

3a  +  5i&  +  2c, 
a  result  which   cannot  be  reduced  to   a   simpler   form. 

33 1  If  some  of  the  quantities  to  be  added  have  similar  terms, 
we  connect  th6  quantities  by  the  sign  of  addition  as  before, 
and  then  reduce  the  resulting  polynomial  to  its  simplest  form, 
by  the  rule  already  given.  This  reduction  will,  in  general,  be 
more  readily  accomplished  if  we  write  down  the  quantities  to 
be  added,  so  that  similar  terms  shall  fall  in  the  same  column. 
Thus; 

Let  it  be  required   to  find    the   sum  of    \    ^  „ 

the   quantities,  / 

^  2ab  —  5^2 

Their  sum,  afler  reducing  (Art.  29),  is     -     5a2  —  6ab  —  4^ 


GHAP    II.]  ADDITION.  27 

34.  As  operations  similar  to  the  above  apply  to  all  algebraio 
expressions,  we  deduce,  for  the  addition  of  algebraic  quantities, 
the   following  general 

RULE. 

L  Write  down  Ike  quantities  to  be  added^  with  their  respective 
signs^  so  that  the  similar  terms  shall  fall  in  the  same  coluinn, 

II.  Reduce  the  similar  terms^  and  annex  to  the  resujts  those  termf 
which  cannot  be  reduced^  giving  to  each  term  its  respective  sign, 

EXAMPLES. 

I.  Add  together  the  polynomials, 

3a2_252-4a6,   ^a'^-b'^  +  2ab   and   3a6  —  Sc^  -  262. 
The  term   Sa^  being  similar   to   Sa^ 
we  write  Sa^  for  the  result  of  the  re- 
duction of  these  two  terms,  at  the  same  < 
time  slightly  crossing   them   as  in  the 
terms  of  the  example. 

Passing  then  to  the  term  —  4a6,  which  is  similar  to  the  two 
terms  +  2a6  and  +  3a6,  the  three  reduce  to  +  a6,  which  is 
placed  after  Sa^,  and  the  terms  crossed  like  the  first  term. 
Passing  then  to  the  terms  involving  6^,  we  find  their  sum  to  be 
—  56^,  after  which  we  write  —  Sc^. 

The  marks  are  drawn  across  the  terms,  that  none  of  them 
may  be  overlooked  and  omitted. 

(2).  (3). 

lx  +  Zab-\'    2c  16a252-f    be  —  2abc 

-^Sx-  Sab  —    5c  —  4a262  —  96c  +  6abc 

5x  —  9ab—    9c  ^  ^  9a262  +    6c  +    a6c 

Sum  .  .   9x  —  9ab  —  12c  3^262  —  76c  +  5a6c 

(4).  (5). 

a  -\-    ab  —    cd+  f  6ab  +  cd  -j-  d 

Sa  +  5a6  —  6cd  ■—  /  3a6  +  ocd  ■—  y 

—  5a  —  6a6  -f  ^cd  —  If  —  4a6  +  6cd  +  x 

-  a+    o.b+    cd  +  4f  ^  f,ab  —  \2cd -\- y 


3^2  _  4j5j5  _  21&2 
5««2  +  2isi6-    &2 
+  3^6  -  2&2 
8a2  ^    ab  —  562  _  3^2 


Sum    —  2a  +    a6 -f  0—3/'  0  0     +  x -{- d 


28  ELEMENTS   OF  ALGEBRA.  [CHAP.   IL 

6.  Add    together    3a  +  b,    Sa -\-  35,     -  9a  —  lb,    6a  +  96   and 
8a  +  36  +  8c.  Ans.  11a  +  96  +8c. 

7.  Add  together   3aa;  4  3ac  +/,    —  9ax  +  7a  +  J,   +  6ax  -j-  3a« 
■f  3/  Sax  +  13ac  +  9/  and  —  14/+  3a:r. 

Ans,  llax  +  19ac  — /+  7a  -f  c?. 

8.  Add  together  the  polynomials,  Sa'^c  -f  5a5,  7a2c  —  3a6  -f  3a€ 
Sa^c  —  6a6  -f-  9ac,  and  —  Sa'^c  -{-  ab  —  12ac,        Ans.  la'^c  —  3a6. 

9.  Add    the    polynomials,     l^aH'^b  —  \2a'^cb,     f)aVb  +  loa^c^ 

—  lOaa;,  —  ^a^x^  —  I'^ahb  and  —  ISaVb  —  12a^cb  +  9  ax. 

Ans.  4a'^x^b  —  22a^cb  —  ax. 

10.  Add   together   3a  +  6  +  c,    5a  +  26  +  3ac,    a -\-  c  -\-  ex   and 

—  3a  —  9ac  ~  86.  Ans.  6a  —  56  +  2c  —  5ac. 

11.  Add    together    5a26  +  Geo;  +  96c2,    7ca;  —  8a26   and    —  15ca; 

—  96c2  +  2a26.  Ans.   —  a%  —  2ca:. 

12.  Add    together    8aa;  +  5a6  +  3a262c2,     -  18aa;  +  Ga^  +  lOaS 
and  lOa^  -  15a6  -  6a26V.  Ans.    —  Za%'^c^  +  6a^. 

13.  What  is  the   sum  of  41a362c  -  27a6c -14a2y   and    10a362r 
+  9a6c'?  Ins.  blaWc  —  18a6c  — 14aY 

14.  "What  is    the    sum   of    18a 6c  —  9a6  +  Gc^  —  3c  +  9aa;   and 
9a6c  +  3c  -  9a^  1  Ans.  27abc  -  9ab  +  6c\ 

15.  What    is    the    sum    of   8a6c  +  b^a  —  2cx  —  6xy    and   7cii 

—  xy  —  1363a?  Ans.  Sabc  —  126%  +  5cx  —  7xy. 

16.  What    is  the    sum   of   9a2c  —  14a6y  +  15a262    and    —  a^c 
— 8a262?  Ans.  Sa^c  -  Uaby -^  7a%\ 

17.  What  is  the   sum  of  17a»62  +  9a36  -  3a2,   -  14a562  +  7c^ 

—  9a3,  -  15a36  +  7a562  -  a^  and  14a36  -  19a36? 

Ans.   . 

18.  What  is  the  sum  of  3aa;2  —  9aa;3  —  17aary,   +  9aa:2  .^  ig^^j 
+  S4:ax7j  and  la^b  +  Sax^  —  7ax^  -f-  46ca;  ?        Ans.  . 

19.  Add    together    3a2  +  5a262c2  —  9a%  7a^  —  8a262c2  —  lOa'^a; 
ttnd  10a6  +  16a262c2  +  19a3a;.  Ans.  lOa^  +  lSa^^c^  +  10a6. 

20.  Add   together    7a26  —  3a6c  —  862c  —  9c3  +  cd^,   Sabc  -  5a26 
4-  3c3  -  462c  +  c(P,  and  4a26  -  8c3  +  962c  -  Sd^ 

Ans.  6a%  +  babe  -  36^0  -14c3  +  2ccr*  -  Sd^ 


CHAP.  II.J  SUBTRACTION.  29 

21.  Add  together  -  ISa^  +  2ab^  +  QaW,  —Sab^ +  la^-5a'^b^ 
and  ~5a36  +  6a5*^hlla262.  Ans.   -- 16a^  +  12aW. 


22.  What    is    the    sum   of   Sa^b^c  —  IGa^a;  —  9ax^d,     +  6a3^>2c 

—  ^ax^d  +  .7a%  and  +  Wax^d  —  a^a:  —  8a362c  1 

Ans,  a%^c  +  aa^^c?. 

23.  What  is  the  sum  of  the  following  terms  :  viz.,  8a^  —  lOa^b 

-  16c^^  +  4a%^  —  12a^b  +  16a^^  +  24.a%^  —  Qab^  —  Wa^^ 
^  20ct253_f.32a5*~855] 

Ans.  8a5  -  22a^b  -  17aW  +  4:8a^P  +  2Qab^  -  86». 


SUBTRACTION. 

35«  Subtraction,  in  algebra,  is  the  operation  for  finding  the 
.fiimplest  expression  for  the  difference  between  two  algebraic 
quantities.     This   difference   is   called   the   remainder. 

36 •  Let  it  be  required  to  subtract  45  from  5a.  Here,  as 
the  quantities  are  not  similar,  their  difference  can  only  be  indi- 
cated, and  we  write 

5a  -  46. 

Again,  let  it  be  required  to  subtract  Aa%   from   la^b.      These 
terms  being  similar,  one  of  them  may  be  taken  from  the  other 
and  their   true   difference   is    expressed '  by 
7a^b  -  4a^  =  Sa^. 

37t  Generally,  if  from  one  polynomial  we  wish  to  subtract 
another,  the  operation  may  be  indicated  by  enclosing  the  second 
in  a  parenthesis,  prefixing  the  minus  sign,  and  then  writing  it 
afler  the  first.  To  deduce  a  rule  for  performing  the  operation 
thus  indicated,  let  us  represent  the  sum  of  all  the  terms  in  the 
first  polynomial  by  a.  Let  c  represent  the  sum  of  all  the  ad- 
ditive terms  in  the  other  polynomial,  and  •—  d,  the  sum  of 
the  subtractive  terms ;  then  this  polynomial  will  be  represented 
by  c  ^  d.     The    operation  may  then  be  indicated  thus, 

a  —  {c    '  d)  'y 
where  it  is  required  to  subtract  from  a  the  difference  between 
€  and  d. 


30  ELEMENTS   OF  ALGEBRA.  ^CHAP.   II. 

• 

If,  now,  we  diminish  the  quantity  a  by  the  quantity  c,  the 
result  a  —  c  will  be  too  smaJ  by  the  quantity  c?,  since  c  should 
have  been  diminished  by  d  before  taking  it  from  a.  Hence, 
to  obtain  the  true  remainder,  we  must  increase  the  first  result 
by  d,  which  gives  the  expression 

'  a  —  c  -i-  d, 

and   this   is   the   true   remainder. 

By  comparing  this  remainder  with  the  given  polynomials,  v^e 
see  that  we  have  changed  the  signs  of  all  the  terms  of  the  quantity 
to  be  subtracted,  and  added  the  result  to  the  other  quantity.  To 
facilitate  the  operation,  similar  quantities  are  written  in  the  same 
column. 

Hence,  for  the  subtraction  of  algebraic  quantities,  we  have  the 
following 

RULE. 

I.  Write  the  quantity  to  be  subtracted  under  that  from  which  it 
is  to  be  taken,  placing  the  similar  terms,  if  there  are  any,  in  the 
same  column, 

II.  Change  the  signs  of  all  the  terms  of  the  quantity  to  be  sub- 
tracted, or  conceive  them  to  be  changed,  and  then  add  the  result  to 
the  other  quantity, 

EXAMPLES. 

(1).       ifi       (1). 

From      -         6ac  —  5ab  +  c^  s'^l         ^clc  —  bab  +  c^ 

Take       -         3ac -f  3a&  —  7c  ^||     —  3ac  —  3a6  +  7c 


Remainder       3ac  —  8a6  +  c^  +  7c.  5  «  I         3ac  —  8a6  +  c2  +  7c. 

,B  o  

(2).  (3). 

From      -         16a2  —    56c  +  lac  19a6c  -—  IQax —    ^axy 

Take       •          14a2  +    5&c  +  8ac  17a6c  +    lax  —  \baxy 

Remamder         2a^  —  \0bc—    ac  .        2abc  —  ^'^ax -\- \Qaxy 

(4).  (5). 

From      -         5a3  — 4a26+    362c  .             4a6  ~    cd+Za^ 

Take       -    -  2^3  _^  3^25  _    352^  5a5  -  4c^  +  ^a^  +  55'-^ 

Remainder       la^ —  la% +  llb'^c  --    a6 -f  3cc^  +  0    -56^ 


CHAP.   II.]  SUBTEACTIOK  31 

6.  From     3a^x  —  ISabc  +  7a^,    take    9a^x  —  ISahc, 

Ans,   —  6a^x  +  7a^. 

7.  From     ^la^^c  -  18a6c  —  Ua^y,      take     41a362c  —  27ab€ 

8.  From     21abc  —  9a5  +  Gc^,    take    9abc  +  3c  —  9aa;.  t 

Ans,  ISabc  —  9a6  +  QC^  —  Sc  +  9ax. 

9.  From     Sabc  —  12i^a  +  5ca;  —  7x2/^   take    7cic  —  xy  —  135%. 

-4/15.  8a6c  +  5%  —  2car  —  Ga-y. 

10.  From   Sa^c  -  14a5y  +  7aW,   take    Oa^c  -  14a5y  +  15a^62^ 

-4/15.   —  a^c  —  8a262. 
'     11.  From    9aV  —  13  +  20a63a;  —  466ca;2,   take    ZbHx^  +  9a6a:2 

-  6  +  Zab'^x,  Ans,  YiabH  -  766ca;2  -  7. 

12.  From    5a*  -  laW  -  ZcH'^  +  7d,    take    3a4  -  3a2  -  IcH-^ 

-  15a352.  ^n5.  2a*  +  Sa^ft^  +  4c5c?2  +  7c?  +  3a2. 

13.  From    51a262  __  48a36  +  10a*,    take    10a*  -  Sa^i  -  Ga^i^. 

Ans,  h7aW  —  40a36. 

14.  From     21:^3^2  +  25a;2y3  +  68^y4  _  49^5^     take     G4a:2?/3 
+  48a://*  —  40y5.  ^Tis.  20a;//*  —  39a:2y3  +  21a;3^2^ 

15.  From     53a:3y2  _  15a;2y3  —  18a:*//  —  5Ga:5,     ^ake     —  \hxHj^ 
+  18a:3y2  +  24a:*//.  ^715.  35a:3//2  _  42^*^  —  5Ga;^ 

38 •  From  what  has  preceded,  we  see  that  polynomials  may  be 
subjected  to  certain  transformations. 

For  example  -     -    -     -     Ga2  —  3a6  +  2^2  _  25c, 
may  be  written  -     -     -     -     ^a?  —  (3a5  —  25^  +  25c). 
In  like  manner  -     -     -     -     7a3  —  8a25  —  45^c  +  65^, 
may  be  written  -    .    .    .    7a3  —  (8a25  +  452c  —  G53)  • 
or,  again,  ......    7a3  —  8a25  —  (452c  —  G53). 

A.lso,     -     - 8a2  -  Ga252  +  5a253, 

becomes 8a2  —  (6a252  —  5a253). 

Also, 9a2c3  —  8a*  +  52  —  c. 

may  be  written ...    -    9a2c3  —  (8a*  —  52  +  c)  ; 
or,  it  may  be  written  -     -     9a2c3  +  52  —  (8a*  +  c). 

These  transformations  consist  in  separating  a  polynomial  into 
two  parts,  and  then  connecting  the  parts  by  the  minus  sign. 


82  ELEMENTS  OF  ALGEBRA.  [CHAP.   IL 

• 

It  will  be  observed  that  the  sign  of  each  term  is  changed  when 
the  term  is  placed  within  the  parenthesis.  Hence,  if  we  have 
one  or  more  terms  included  within  a  parenthesis  having  the 
minus  sign  before  it,  the  signs  of  all  the  terms  must  he  changed 
when  the  parenthesis  is  omitted. 

ThuS;  4a  —  (6a5  —  3c  —  26), 

is  equal  to  .        4a  —   ^ah  +  3c  +  26. 

Also,  Qah  —  {—  4:ac  -\- Sd  —  4a6), 

is  equal  to  6ab     +     4ac  —  3o?  -f  4a6. 

39.  Remark. — From  what  has  been  shown  In  addition  and 
subtraction,  we  deduce  the  following  principles. 

1st.  In  Algebra,  the  words  add  and  sum  do  not  always,  as  in 
arithmetic,  convey  the  idea  of  augmentation.  For,  if  to  a  we  add 
—  .6,  the  sum  is  expressed  by  a  ~  b,  and  this  is,  properly  speaking, 
the  arithmetical  difference  between  the  number  of  units  expressed 
by  a,  and  the  number  of  units  expressed  by  6.  Consequently, 
this  result  is  actually  less  than  a. 

To  distinguish  this  sum  from  an  arithmetical  sum,  it  is  called 
the  algebraic  sum. 

Thus,  the  polynomial,     2a^  —  Sa^  -j-  Sb^c, 
is  an  algebraic  sum,  so  long  as  it  is  considered  as  the  result  of 
the  union  of  the  monomials 

2a3,    -  3a26,    +  ^b\ 
with    their    respective   signs;    but,  in  its  proper  acceptation^  it  is 
the    arithmetical    difference    between   the   sum  of  the   units   con- 
fined  in    the    additive   terms,    and    the   units    contained   in   the 
subti^active  term. 

It  follows  from  this,  that  an  algebraic  sum  may,  in  the  numer 
ical  applications,  be  reduced  to  a  negative  expression. 

2d.  The  words  subtraction  and  dij^rence,  do  not  always  convey 

the  idea  of  diminution.      For,   the   difference   between    -f  a  and 

—  b   being 

a  —  (—  b)  =  a  +  b, 

is  numerically  greater  than  a.     This  result  is  an  algebraic  differ 
ence. 


CHAP.  11.1  MULTIPLICATION  83 

40.  It  frequently  occurs  in  Algebra,  that  the  algebraic  sign  -|- 
or  — ,  which  is  written,  is  not  the  true  sign  of  the  teim  before 
wliich  it  is  placed.  Thus,  if  it  were  required  to  subtract  —  h 
from  a,  we  should  write 

a  —  (  —  5)  =  a+6. 
Here  the  true  sign  of  the  second  term  of  the  binomial  is  plus, 
alf>hough  its  algebraic  sign  is  — .  This  minus  sign,  operating 
upon  the  sign  of  5,  which  is  also  negative,  produces  a  plus  sign 
for  b  in  the  result.  The  sign  which  results,  after  combining  the 
algebraic  sign  with  the  sign  of  the  quantity,  is  called  the  esseri- 
tial  sign  of  the  term^  and  is  often  different  from  the  algebraic 
sign. 


MULTIPLICATION. 

41  •  Multiplication,  in  Algebra,  is  the  operation  of  finding  the 
product  of  two  algebraic  quantities.  The  quantity  to  be  multi- 
plied is  called  the  multiplicand ;  the  quantity  by  which  it  is 
multiplied  is  called  the  multiplier ;  and  both  are  called  factors, 

42.  Let  us  first  consider  the  case  in  which  both  factors  are 
monomials. 

Let  it  be  required  to  multiply  7aW  by  4a'^b  ;  the  operation 
may  be  indicated  thus, 

7a362  X  4a26, 
or    by    resolving    both    multiplicand    and    multiplier    into    their 
simple  factors,  ' 

laaabb  X  4aa6. 

Now,  it  has  been  shown  in  arithmetic,  that  the  value  of  a 
product  is  not  changed  by  changing  the  order  of  .ts  factors; 
hence,  we  may  write  the  product  as  follows: 

7  X  Aaaauahbb,  which  is  equivalent  to  28a^P. 
Comparing  this  result  with  the  given  factors,  we  see  that  die 
CO  efficient  in  the  product  is  equal  to  the  product  of  the  co-effi- 
cients of  the  multiplicand  and  multiplier ;  and  that  the  exponent 
of  each  letter  is  equal  to  the  sum  of  the  exponents  of  that  letter 
'  ..  both  multiplicand  and  multiplier. 


84  ELEMENTS   OF  ALGEBRA.  [CHAP.    IL 

And  since  the  sam^  course  of  reasoning  may  be  applied  to 
anv  two  monomialsj  we  have,  for  the  multiplication  of  mono 
inials,  the  following 

RULE. 

I,  Mt^ltiph,   the  co-efficients  together  for  a  new  co-efficient. 

II.  Write  after  this  co-efficient  all  the  letters  which  enter  into  th$ 
multiplicand  and  multiplier,  giving  to  each  an  exponent  equal  to 
the  sum  of  its  exponents  in  both  factors. 

EXAMPLES. 

(1)       .       -       8a25c2    X  7a5d:2  =  ^aWc'^d'^. 
^2)       -       -     2laWdc  X  8a5c3  =  IGSa^SVd 

(3)  (4)  (5)  (6) 

Multiply-       -       3a26  -     \2a'^x    -    -     Qxyz         -       a^xy 

by        -       -       2ha^     -     -     Ylx'^-y     -     -       ay'^z    -     -     ^xy"^ 


6a^b^  14Aa^x'^y  Qaxy^z"^  2aVy^. 

7.  Multiply    Sa^b^c  by   7a%^cd.  Ans.  66a^%^c''d. 

8.  Multiply    5abd^   by    12cd^.  Ans.  QOabcd^ 

9.  Multiply    la'^bd'^c^  by   abdc.  Ans.  7a^b^d^cK 

43.  We  will  now  proceed  to  the  multiplication  of  polynomials. 
In  order  to  explain  the  most  general  case,  we  will  suppose  the 
multiplicand  and  multiplier  each  to  contain  additive  and  sub- 
tractive   terms. 

Let  a  represent  the  sum  of  all  the  additive  terms  of  the  multi- 
plicand, and  —  b  the  sum  of  the  subtractive  terms ;  c  the  sum 
of  the  additive  terms  of  the  multiplier,  and  —  d  the  sum  of 
the  subtractive  terms.  The  multiplicand  will  then  be  represented 
by  a  —  5   and  the  multiplier,  by   c  —  d. 

We  will  now  show  how  the  multiplication  expressed  by 
(a  —  6)  X  (c  —  d)   can   be   effected. 

The  required  product  is  equal  to   a  —  6 

taken   as   many   times    as    there  are  units 

in    c  —  d.      Let   us   first    multiply  by   c  ; 

.  <^c  —  bc 
that    IS,   take    a  —  o    as    many    times    as  ,       ,  , 

___    _      .  .  —  art  +  bd 

there  are  units  in   c.     We  begm  by  writ- ; :; ~ 

ac  —  be  — -  ad  -f-  bd* 
ing  ac,  which    is    too  great    by  b    taken 


CHAF.   II.]  MULTIPLICATION,  85 

c  times ;  for  it  is  only  the  difference  between  a  and  6,  that  is 
first  to  be  multiplied  by  c.  Hence,  ac  —  be  is  the  product  of 
a  —  b   by   c. 

But  the  true  product  is  a  —  6  taken  c  —  d  times :  hence,  the 
last  product  is  too  great  by  a  —  6  taken  d  times  ;  that  is,  by 
gd  —  bd^  which  must,  therefore,  be  subtracted.  Suotracting  this 
from  the  first  product  (Art.  37),  we  have 

{a  —  b)  X  {c  —  d)  —  ac  —  be  —  ad  +  bd : 

If  we  suppose  a  and  c  each  equal  to  0,  the  product  will  re 
duce  to  i-  bd, 

44»  By  considering  the  product  of  a  -—  6  by  c  —  df,  we  may 
deduce  the  following  rule  for  signs,  in  multiplication. 

When  two  terms  of  the  multiplicand  and  multiplier  are  affected 
with  the  same  sign,  their  product  will  be  affected  with  the  sign  -f, 
and  when  they  are  affected  with  contrary  signs,  their  product  will 
be  affected  with  the  sign  — . 

We  say,  in  algebraic  language,  that  +  multiplied  by  -f 
or  —  multiplied  by  — ,  gives  -f-  ;  —  multiplied  by  +,  or  +  mul 
tiplied  by  — ,  gives  — .  But  since  mere  signs  cannot  be  multi- 
plied together,  this  last  enunciation  does  not,  in  itself,  express  a 
distinct  idea,  and  should  only  be  considered  as  an  abbreviation 
of  the  preceding. 

This  is  not  the  only  case  in  which  algebraists,  for  the  sake  of 
brevity,  employ  expressions  in  a  technical  sense  in  order  to  se- 
cure  the  advantage  of  fixing   the   rules   in   the   memory. 

45.  We  have,  then,  for  the  multiplication  of  polynomials,  the 
following 

RULE. 

Multiply  all  the  terms  of  the  multiplicand  by  each  term  of  the 
multiplier  in  succession,  affecting  the  product  of  any  two  terms  with 
the  sign  plus,  when  their  signs  are  alike,  and  with  the  sign  minus, 
when  their  signs  are  unlike.  Then  reduce  the  polynomial  result 
to  its  sim^ilest  form. 


OO  ELEMENTS  OF  ALGEBRA.   ^  I  CHAP.   IL 

EXAMPLES. 

1.  Multiplj 3a2  +  4a6  +  6« 

ly 2a   +  55 

6a3  +    Sa^  +    2ab'^ 

+  15a^  +  20ab^  +  5&* 
Product  -  -  -.   6a3  -f  23a26  +  22ab^  +  5^3 

(2).  (3). 

a:^  -I-  2/2  ^5  ^  ^^6  _j_  y-jfax 

X   —  y  ax  -{-  5ax 

x^  -f-  ^y^  «^^  +  «^^3/^  +  la'^x'^ 

—  x'^y  —  y^  +  5ax^   +  bax'^y^  -f-  35a^a;^ 

a;^  +  iry^  —  x'^y  —  y"^  Qax^  -\-  (jax^y^  +  4i2a'^x'^. 

4.  Multiply   x^  +  2ax  -{-  a^  by   x  -\-  a. 

Arts,  x^  +  Sax'^  +  3a2a;  +  a^^ 

5.  Multiply   a;2  -|-  y2   i3y   ^  _|_  ^^ 

^W5.  ic^  +  ^y^  +  x^y  +  y^. 

6.  Multiply   3a62  +  Ga^c^   by   Sab^  +  3a2c2. 

^^5.  9^264  +  27a^^c^  +  18aM. 

7.  Multiply   4a;2  —  2y   by   2y.  ^7i5.  8a;2?/  —  4y2. 

8.  Multiply   2a;  +  4y   by   2x  —  4y.  Ans,  4a;2  —  16y2. 

9.  Multiply   x^  +  a;2y  +  a??/2  -\-  y^   by   x  —'  y.         Ans.  . 

10.  Multiply   x"^  +  xy  +  y'^   by   x^  —  xy  -\'  2/2. 

u4;z5.  a;*  +  ^^y^  +  y** 
In  order  to  bring  together  the  similar  terms,  in  the  product  o 
two  polynomials,  we  arrange  the  terms  of  each  polynomial  "witQ 
reference  to  a  particular  letter ;  that  is,  we  arrange  them  so  tha 
tlie  exponents  of  that  letter   shall   go    on   diminishing  from  left 
to   right. 

11    Multiply    4a3~    '5a25  -    ^ab'^   +  2Z>3 

by  2a2  —    3a5    —    4^2 

8a5  -  lOa^i  -  }6a362  +    4a253 
J  -  12a*6  +  15a^62  +  24a2^.3 -.    6a5* 

!  __^ -  \%aW  +  20^253  4-  32a5^  ~  85« 

8a5  _  22a*5  -  17a362  +  48a263  -j.  ^^ab^  _  85^.  • 


CHAP.  II.J  MULTIPLICATION.  $7 

After  having  arranged  the  polynomials,  with  reference  to  the 
letter  a,  multiply  each  term  of  the  first,  by  the  term  2a^  of  the 
second ;  this  gives  the  polynomial  SaP  —  lOa'^b  —  l^a^"^  +  Aa^P, 
in  which  the  signs  of  the  terms  are  the  same  as  in  the  multi- 
plicand. Passing  then  to  the  term  —-  Sab  of  the  multiplier,  muL 
tiply  each  term  of  the  multiplicand  by  it,  and  as  it  is  affected 
with  the  sign  — ,  affect  each  product  with  a  sign  contrary  to 
that  of  the  corresponding  term   in  the  multiplicand ;   this   gives 

—  12a^b  +  15a362  +  24a^^  -  6ab\ 
Multiplying   the  multiplicand  by    —  Ab^,   gives 
—  16a362  +  20a263  +  ^2ab^  —  Sb\ 

The  product  is  then  reduced,  and  we  finally  obtain,  for  the  most 
Bimple  expression   of  the  product, 

Sa^  -  22a^b  -  17a%^  +  4:Sa^^  +  26ab^  ~  Sb^ 

12.  Multiply   2a2  —  Sax  +  4:X^  by   5a2  _  6ax  —  2x\ 

Ans.  10a*  —  27a^x  +  S4a'^x^  —  18ax^  —  Sx^. 

13.  Multiply   3a;2  — 2ya;  +  5   by   x^  +  2x9/  —  S. 

Ans.  Sx^  +  4x^7/  —  4x^  —  4x'^7/^  +  16x9/ —  lb. 

14.  Multiply   3^3  _^  2xh/  +  Sy^  by   2x^  —  Sxhf  +  bi/, 
\6x^  —  5a;V  —  6:rV*  +  6a;3y2  4.  15a;3y« 


•  9a;2y4  +  \{^xhf  +  15?/5. 

15.  Multiply   8ax  —  6ab  —  c  by   2ax  +  a6  +  c. 

Ans,  16a2a;2  —  4a^bx  —  6aW  +  Qacx  —  labc  —  c^, 

16.  Multiply   3a2  —  5^2  _|.  3^2   \yj   a^  —  b\ 

Ans.  3a*  —  5a262  +  Sa^c^  —  3a263  +  5b^  —  35V. 

17.  Multiply   3a2  -  5bd  +    cf 
by  -  5a2  +  4bd  -  Sc/. 

Product  —  15a*  ■j-'^a'^bd  —  29a'^cf  —  20b^d^  +  AUcd/--  sJ^/^. 

18.  Multiply   4a'^52  —  5a^^c  +  8a25c2  -  3a2c3  —  7abc^ 
by  2a62   ^  4a6c    ~-2bc^     +  c^. 

r  8a*5*     —  lOa^Mc  +  2Sa^h^  —  S4aWc^ 

rroduct    I  —    4a263c3  —  IGa^^^c  +  12a36c*  +  7a252c* 
(   +  Ua^c^  +  14a62c5  —    3a2c6     ~  7a5c«. 


38  ELEMENTS   OF  ALGEBRA.  [CHAP.   IL 

46  •      REMARKS    ON    THE    MULTIPLICATION    OF    POLYNOMIALS. 

Ist.  If  both  multiplicand  and  multiplier  are  homogeneous^  the 
product  will  be  homogeneous^  and  the  degree  of  any  term  of  the 
product  ivill  he  indicated  by  the  sum  of  the  numbers  which  iidicate 
the  degrees  of  its  two  factors. 

Thus,  in  example  18th,  each  term  of  the  multiplicand  is  of 
the  5th  degree,  and  each  term  of  the  multiplier  of  the  3d  de- 
gree :  hence,  each  term  of  the  product  is  of  the  8th  degree. 
This  remark  serves  to  discover  any  errors  in  the  addition  of 
the   exponents. 

2d.  If  no  two  terms  of  the  product  are  similar^  there  will  be  no 
reduction  amongst  them ;  and  the  number  of  terms  in  the  product 
ivill  then  be  equal  to  the  number  of  terms  in  the  multiplicand^  multi 
plied  by  the  number  of  terms  in  the  multiplier. 

This  is  evident,  since  each  term  of  the  multiplier  will  produce 
as  many  terms  as  there  are  terms  in  the  multiplicand.  Thus,  in 
example  16th,  there  are  three  terms  in  the  multiplicand  and  two 
in  the  multiplier :  hence,  the  number  of  terms  in  the  p~'o<^uct  is 
equal  to   3  X  2  =:  6. 

Sd.  Among  the  terms  of  the  product  there  are  always  two  which 
cannot  be  reduced  with  any  others. 

For,  let  us  consider  the  product  with  reference  to  any  letter 
common  to  the  multiplicand  and  multip'.'er :  Then  the  irreduci- 
ble  terms   are, 

1st.  The  term  produced  by  the  multiplication  of  the  two  terms 
of  the  multiplicand  and  multiplier  which  contain  the  highest 
power   of  this   letter ;    and 

2d.  The  term  produced  by  the '  multiplication  of  the  two  terms 
which   contain   the   lowest   power  of  this   letter. 

For,  these  two  partial  products  will  contain  this  letter,  to  a 
higher  and  to  a  lower  power  than  either  of  the  other  partial  pro 
ducts,  and  consequently,  they  cannot  be  similar  to  any  of  them. 
This  remark,  the  truth  of  which  is  deduced  from  the  law  of 
th«^-   exponents,  will   be  very  useful   in   division. 


CHAP.   11.]  MULTIPLICATION.  89 

EXAMPLE. 

Multiply     -     .      5a*52  +  3^25  _  ^54  _  2ab^ 
by      -      -      -        a^b   —    a^^ 

^^^^^^  I  -  6a'b^  -  3a363  4,    ^256  4.  2a^\ 

If  we  examine  the  multiplicand  and  multiplier,  with  reference 
to  a,  we  see  that  the  product  of  5a*52  }yj  g2jy^  must  be  irre- 
ducible ;  also,  the  product  of  —  2^6^  by  ab'^.  If  we  consider 
the  letter  5,  we  see  that  the  product  of  —  a6^  by  —  aS^,  must 
be  irreducible,  also   that   of  ^a^b   by   a^b, 

47«  The  following  formulas  depending  upon  the  rule  for  mul- 
tiplication, will  be  found  useful  in  the  practical  operations  of 
algebra. 

Let  a  and  b  represent  any  two  quantities ;  then  a  +  b  will 
represent  their   sum,  and  a  —  b   their  difference. 

I.  We   have       {a  +  bf  =z  {a  +  b)  X  {a  +  b), 
or   performing  the   multiplication  indicated, 

(a  +  by  =za^  -^  2ab  +  b^ ;  that  is. 
The  square  of  the  sum  of  two  quantities  is  equal  to  the  square 

of  the  first ^  jp^us  twice  the  product  of  the  fij'st  by  the  second^  plus 

the  square  of  the  second. 

To   apply  this  formula  to  finding   the   square  of  the  binomial 

we   have      (5a2    +  ^a%Y  =  25a*     +    SOa^J   +  ^Aa^b"^. 
Also,  {Qa'^b  +  9a&3)2  ~  SGa^is  +  lOSa^J*  +  ^la^¥. 

II.  We  have,      {a  -  bf  =  {a  -  b)  x  {a  ^  b\ 
or  performing   the  multiplication  indicated, 

{a  -  by  =  a'-2ab  +  b'''^  that  is. 
The  square  of  the  difference   between  two  quantities  is  equal   to 
the  square  of  the  first,  minus  twice  the  product  of  the  first  by  the 
seco7id,  plus  the  square  of  the  second. 

To  apply  this   to   an   example,  we    nave 

(7a2i2  .    i2ab^Y  =  49a^b^  -  l6Sa^^  +  U4a^\ 
Also,        (4^353        7cV3)2  =  I6a%^  —    66a^^c^d^  +  4:9c^d\ 


4C  ELEMENTS   OF  ALGEBRA.  [CHAP.   IL 

IIJ.  We  have       (a  +  b)  X  {a  -  b)  =  a?  -  b\ 
by  performing   the   multiplication ;    that  is, 

The  sum  of  two  quantities  multiplied  by  their  difference  is  equal 
to  the  difference  of  their  squares. 

To   apply  this  formula   to   an   example,  we   have 

(8a3   +  7a62)  x  (Sa^   -  lab"^)  =  64a6    -  4:9a'^¥, 

48t  By  considering  the  last  three  results,  it  is  perceived 
tiiat  their  composition,  or  the  manner  in  which  they  are  formed 
from  the  multiplicand  and  multiplier,  is  entirely  independent  of 
Mij  particular  values  that  may  be  attributed  to  the  letters  a  and 
i,  which  enter   the   two  factors. 

The  manner  in  which  an  algebraic  product  is  formed  from  its 
two  factors,  is  called  the  law  of  the  product ;  .and  this  law  re- 
mains always  the  same,  whatever  values  may  be  attributed  to 
the   letters  which  enter   into   the  two  factors. 


DIVISION. 

49*  Division,  in  algebra,  is  the  operation  for  finding  from  two 
given  quantities,  a  third  quantity,  which  multiplied  by  the  second 
shall   produce   the  first. 

The  first  quantity  is  called  the  dividend^  the  second^  the  divisor^ 
and   the  third^  or   the   quantity  sought,  the  quotient. 

50»  It  was  shown  in  multiplication  that  the  product  of  two 
terms  having  the  same  sign,  must  have  the  sign  +?  and  that 
the  product  of  two  terms  having  unlike  signs  must  have  the 
sign  — .  Now,  since  the  quotient  must  hav^  such  a  sign  that 
when  multiplied  by  the  divisor  the  product  will  have  the  sign  of 
the   dividend,  we  have  the  following  rule  for  signs   in  division. 

If  the  dividend  is  +  and  the  divisor  -r  the  quotient  is  -f-  ; 
if  the  dividend  is  -{-  and  the  divisor  —  the  quotient  is  —  ; 
if  the  dividend  is  —  and  the  divisor  +  the  quotient  is  —  ; 
if  the    dividend  is    —  and   the    divisor    —    the  quotient  is    -(-. 

That  is  :  The  quotient  of  terms  having  like  signs  is  plus,  ana 
the  quotient  of  terms  having  unlike  signs  is  minims. 


CHAP.   II.j  DIYISIOK.  41 

61.  Let  us  first  consider  the  case  in  which  both  dividend  and 
divisor  are  monomials.     Take 

35a^6V   to   be,  divided  by   laP-hc, 

The  operation  may  be  indicated  thus, 
35a5^.2,2 

-  ^  ^, —  :    quotient,  ba^c. 
7aHc 

Ncvr,  since  the  quotient  must  be  such  a  quantity  as  multiplied 
by  the  divisor  will  produce  thr  dividend,  the  co-efficient  of  the 
quotient  multiplied   by  7   mus\    give  35  ;   hence,  it  is  5. 

Again,  the  exponent  of  each  ]  etter  in  tie  quotient  must  be  such 
that  when  added  to  the  exponent  of  the  same  letter  in  the  divisor, 
the  sum  will  be  the  exponent  of  that  letter  in  the  dividend. 
Hence,  the  exponent  of  a  in  the  quotient  is  3,  the  exponent  of 
5  is   1,   that  of  c  is  1,  and   the   required  •  quotient  is   6a^c, 

Since  we  may  reason  in  a  similar  manner  upon  any  two 
monomials,  we  have  for  the  division  of  monomials  the  following 

RULE. 

I.  Divide  the  co-efficient  of  the  dividend  by  the  co-efficient  of  the 
divisor^  for  a  new  co-efficient, 

II.  Write  after  this  co-efficient,  all  the  letters  of  the  dividend 
and  give  to  each  an  exponent  equal  to  the  excess  of  its  expo 
nent  in    the   dividend  over   that  in    the  divisor. 

By  this   rule  we  find, 

A^aWchl       ,  „  ^  '   150a5S3cc?3 

EXAMPLES. 


1. 

Divid 

16a;2  by   8rr. 

Ans.  2x. 

2. 

Divide 

l^a^xy^   by   3ay. 

Ans.  5axy^ 

S. 

Divide 

Mah^x  by   1262. 

Ans,  lahx. 

4. 

Divide 

~96a4Z^2^3   by    l^a^c. 

Ans,   -Sa^c\ 

5. 

Divide 

144a95V^5   i3y    ^^Qa^h^c^d, 

Ans.   —4:a^b'^cdK 

6. 

Divide 

-- 256a35c2a;3   by    —  16a2ca;2. 

Ans,  \Qabcx. 

7. 

Divide 

—  mOa^h^c^x'^   by   ZOa^hh'^x. 

Ans,   —  \Oabcx, 

8 

Divide 

—  400a856c*^5   bv   25a86Wtr. 

Ans.    -.166c.r*. 

42  ELEMENTS  OF  ALGEBKA.  [CHAP.    II. 

62'»  It  follows  from  the  preceding  rule  that  the  exact  division 
of  monomials  will   be  impossible  : 

1st.  When  the  cc-efficient  of  the  dividend  is  not  divisible  by 
that   of  the   divisor. 

2d.  When  the  exponent  of  the  same  letter  is  greater  in  the 
divisor   than  in   the   dividend. 

This  last  exception  includes,  as  we  shall  presently  see,  the 
case  in  which  the  divisor  has  a  letter  which  is  not  contained 
in  the  dividend. 

When  either  of  these  cases  occurs,  the  quotient  remains  un- 
der the  form  of  a  monomial  fraction  ;  that  is,  a  monomial 
expression,  necessarily  containing  the  algebraic  sign  of  division. 
Such   expressions   may  frequently  be   reduced. 

Take,  for   example,     -g^-  ^  -^. 

Here,  an  entire  monomial  cannot  be  obtained  for  a  quotient; 
for,  12  is  not  divisible  by  8,  and  moreover,  the  exponent  of  c 
is  less  in  the .  dividend  than  in  the  divisor.  But  the  expression 
can  be  reduced,  by  dividing  the  numerator  and  denominator  by 
the  factors  4,  o?^  b,  and  c,  which  are  common  to  both  terms 
of  the  fraction. 

In  general,  to  reduce  a  monomial  fraction  to  its  lowest  terms: 

Suppress  all  the  factors  common  to  both  numerator  and  denomi- 
nator. 

From   this   rule  we  find, 

ASaWcd^        4ad^  .  Slabh'^d       S7b^c 


^^^^'  i\^3L^A^2  —  ' 


SQa^^chle  ~  Zbce  '  '  6a^c^d^  ~   QaH  ' 

\2a%^c'  _  Sab  _7a^b_  _  _1_ 

'         16a^5^  ~  4^'  '        Ua^  ~  2ab  ' 

In  the  last  example,  as  all  the  factors  of  the  dividend  are 
found  in  the  divisor,  the  numerator  is  reduced  to  1  ;  for,  in  fact, 
both   terms   of  the  fraction  are  divisible   by  the  numerator*. 

53,  It  often  happens,   that    the    exponents  of  certain  letters, 

are  the   surae  in   the   dividend  and  divisor. 

24^362 
i^or  example,    -    -    -    -     — — , 


CHAP.   II.J  DIVISION.  43 

is  a  case  ii3  which  the  letter  h  is  affected  with  the  same  expo- 
nent in  the  dividend  and  divisor :  hence,  it  will  divide  out,  and 
will   not  appear   in   the   quotient. 

But  if  it  is  desirable  to  preserve  the  trace  of  this  letter  in 
the  quotient,  we  may  apply  to  it  the  rule  for  exponents  (Art. 
51),  which  gives 

62 

-  =  62-2  =  60. 

6^ 

The   symbol   6®,  indicates  that  the   letter  6  enters   0  times  as 

A  factor  in  the  quotient  (Art.   16) ;  or  what  is  the  same   things 

that  it  does  not  enter  it  at  all.     Still,  the  notation  shows  that  6 

was  in  the  dividend  and   divisor  with   the   same   exponent,  and 

has   disappeared  by   division. 

1 5ft26  c2 
In   like   maimer,        ^  m  ^    =  ^oPh\^  =  562. 
3a26c2 

54 •  We  will  now  show  that  the  power  of  any  quantity  whose 
exponent  is  0,  is  equal  to  1.  Let  the  quantity  be  represented 
by  a,  and   let   m   denote   any  exponent  whatever. 

Then,      —  =  a""""  =  a°,   by  the  rule  for  division. 

But,         —  =  1,  since  the  numerator  and  denominator  are  equal : 

nence,         a^  =  1,  since  each  is  equal  to  — 

We  observe  again,  that  the  symbol  a^  is  only  employed  con- 
ventionally, to  preserve  in  the  calculation  the  trace  of  a  letter 
which  entered  in  the  enunciation  of  a  question,  but  which  may 
disappear   by  division. 

55#  In  the  second  place,  if  the  dividend  is  a  polynomial  and 
the  divisor  is  a  monomial,  we  divide  each  term  of  the  dividend 
by  the  divisor y  and  connect  the  quotients  by  their  respective  signs, 

EXAMPLES. 

Divide    Qa^x^y^  —  X^a^x^y^  +  Iba^a^y^    by    Sa'^x^y^, 

Ans.  2x^y^  *-  Aaxy^  +  ia^x^y. 


44  ELEMENTS  OF  ALGEBRA.  |CHAP.   XL 

Divide    12a^y^  —  lQa^7/  +  20a6y*  —  28a>3    by    —  4a^y3. 

Ans,   —  32/3  +  4ay2  __  5^2^  _|_  7^3, 

Divide    ISa^Sc  —  20ac7/^  +  6cd^    by  —  babe, 

.  ^     ,    4?/2       d?" 

0        ah 

56 1  In  the  third  place,  when  both  dividend  and  divisor  are 
polynomials.      As    an    example,    let    it    be    required    to    divide 

26^252  _|.  lOa*  —  48a36  +  24.ob^    by   4a&  -  ^o?  +  3^2. 
In    order   that  we   may  follow  the   steps  of  the   operation  more 
easily,  we  will  arrange  the  quantities  with  reference  to  the  letter  a. 

Dividend,  Divisor, 

lOa^  -  48a36  +  26a2^»2  4.  2^a¥  j  |  -  5a2 -f  4a5  +  3^2 

It  follows  from  the  definition  of  division  and  the  rule  for  the 
multiplication  of  polynomials  (Art.  45),  that  the  dividend  is  the 
sum  of  the  products  arising  from  multiplying  each  term  of 
the  divisor  by  each  term  of  the  quotient  sought.  Hence 
if  we  could  discover  a  term  in  the  dividend  which  was  derived, 
without  reduction,  from  the  multiplication  of  a  term  of  the  divi 
sor  by  a  term  of  the  quotient,  then,  by  dividing  this  term  </ 
the  dividend  by  that  term  of  the  divisor,  we  should  obtain  one 
term  of  the   required   quotient. 

Now,  from  the  third  remark  of  Art.  46,  the  term  10a*,  con 
taining  the  highest  power  of  the  letter  a,  is  derived,  without 
reduction  from  the  two  terms  of  the  divisor  and  quotient,  con- 
taining the  highest  power  of  the  same  letter.  Hence,  by  dividing 
the  term  10a*  by  the  term  —  5a2,  we  shall  have  one  term  of 
the   required   quotient. 

Dividend,  Divisor, 


10a*  ~  48(^36  +  26a2i2  +  24.aP 
f  10a*  -    8a35  -    6a262 


l-5a2  +  4a6  +  362 


-  2a2  +  8a6 
40a36  +  32a2Z>2  +  24a^3  Quotient, 

40a35  +  32a262  _|_  24a63. 


Since  the  terms  10a*  and  —  5a2  are  iffected  with  contrarj^ 
signs,  their  quotient  will  have  the  sign  —  ;  hence,  10a*,  divided 
by  —  5a2,  gives  —  2a2  for  a  .erm  of  the   required   quotient. 


CHAP.   II.]  DIVISION.  45 

After  having  written  this  term  under  the  divisor,  multiply  each 
term  of  the  divisor  by  it,  and  subtract  the  product, 

from   the   dividend.     The  remainder  after  the  first  operation  is 
—  40a36  +  32a262  +  24abK 

This  result  is  composed  of  the  products  of  each  term  of  the 
divisor,  by  all  the  terms  of  the  quotient  which  remain  to  be 
determined.  We  may  then  consider  it  as  a  new  dividend,  and 
reason  upon  it  as  upon  the  proposed  dividend.  We  will  there- 
fore divide  the  term  —  40a^,  which  contains  the  highest  power 
of  a,  by  the  term  —  5a^  of  the  divisor. 

This  gives  -{-Sab 

for  a  new  term  of  the  quotient,  which  is  written  on  the  right 
of  the  first.  Multiplying  each  term  of  the  divisor  by  this  term 
of  the  quotient,  and  writing  the  products  underneath  the  second 
dividend,  and  making  the  subtraction,  we  find  that  nothing  re- 
mains.    Hence, 

—  2a2  +  Sab    or    Sab  —  2a^ 
is  the  required  quotient,  and  if  the  divisor  be  multiplied  by  it, 
the  product  will  be  the  given  dividend. 

By  considering  the  preceding  reasoning,  we  see  that,  in  each 
operation,  we  divide  that  term  of  the  dividend  which  contains 
the  highest  power  of  one  of  the  letters,  by  that  term  of- the 
divisor  containing  the  highest  power  of  the  same  letter.  Now, 
we  avoid  the  trouble  of  looking  out  these  terms  by  arranging 
both  polynomials  with  reference  to  a  certain  letter  (Art.  45), 
which  is   then   calkd   the  leading  letter. 

Since  a  similar  course  of  reasoning  may  be  had  upon  any  two 
polynomials,  we  have  for  the  division  of  polynomials  the  following 

RULE. 

I.  Arrange  the  dividend  and  divisor  with  reference  to  a  certain 
letter^  and  then  divide  the  first  term  on  the  left  of  the  dlvilend  by 
the  first  term  on  the  left  of  the  divisor,  for  the  first  term  of  the 
quotient ;  multi'ply  the  divisor  by  this  term  and  subtract  the  pro- 
duct from  the  dividend. 


46  ELEMENTS   OF  ALGEBRiiR  [CHAP.   II, 

II.  Then  divide  the  first  term  of  the  remainder  hy  the  first  term 
of  the  divisor,  for  the  second  term  of  the  quotient ;  multiply  the 
divisor  hy  this  second  term,  and  subtract  the  'product  from  the 
result  of  the  first  operation.  Continue  the  same  operation  until  a 
remainder  is  found  equal  to  0,  or  till  the  first  term  of  the  remainder 
is  not  exactly  divisible  by  the  first  term  of  the  divisor. 

In  the  first  case,  (that  is,  when  the  remainder  is  0,)  the 
division  is  said  to  be  exact.  In  the  second  case  the  exact  divi- 
sion  cannot  be  performed,  and  the  quotient  is  expressed  by 
writing  the  entire  part  obtained,  and  after  it  the  remainder  with 
its   proper  sign,  divided  by  the  divisor. 

SECOND    EXAMPLE. 

Divide      ^Ix^y"^  +  2bx'^y^  +  ^Sxy^  —  40y5  —  56^^  _  ig^i^     i^y 

^y1  -_  8^2  __  g^y^ 

—  40y5  -f.  68^?/4  +  25a;2y^  +  21a:V  _  13^4^  _  56a;5||5?/2  —  6a:y-8a;2 

1st  rem.  20^y*  —  39^2^^  _|_  21rrV 
20:ry*  —  24aj2?/3  _  822^3^-2 

2d  rem.       -       —  X^x^y"^  -^  ^Sx'^y'^  —  ISx'^y 
—  Ibx^y-  +  18^3?/2  +  24^  V 


8d.  rem.      -      -       -      -      S5x^y^  —  A2x^y  —  ^x^ 

35a;  V  —4,2x^y  —  b^x^ 

Final  remainder 0. 

67.  Remark. — In  performing  the  division,  it  is  not  necessary 
to  bring  down  all  the  terms  of  the  dividend  to  form  the  first 
remainder,  but  they  may  be  brought  down  in  succession,  as  in 
the  example. 

As  it  is  important  that  beginners  should  render  themselves 
(jimiliar  with  algebraic  operations,  and  acquire  the  habit  of  calcu- 
lating promptly,  we  will  treat  this  last  example  in  a  different 
manner,  at  the  same  time,  indicating  the  simplifications  which 
should  be  introduced.  These  consist  in  subtracting  each  partiaJ 
product  from  the  dividend  as  soon  as  this  product  is  formed. 


CHAP.   II.]  DIVISIOM  47 

—  40y5  +  eSxy*  +  25x^f  +  2lx^y^  —  ISx^y  —  ^6x^\  |5y2  —  Grry  —  Sx^ 

1st  rem.    20;ry^  —  39^V  +  21:r V  —  Sy^-{-  Axy"^  —  3:^2^  ^7^4 

2d  rem.       -       —  Ibx'^y^  +  SS^^y^  —  18ic*y 

3d   rem.       -       .       -       -      35a; V  —  42^;^  —  56a;« 

Final  remainder       -  -      -        0. 

First,  by  dividing  —  40?/^  by  S?/^,  we  obtain  —  8y^  for  Ihe 
quotient.  Multiplying  Sy^  by  —  Sy^,  we  have  —  40?/^,  or,  by 
changing  the  sign,  +  40y^,  which  cancels  the  first  term  of  the 
dividend. 

In  like  manner,  —  Qxy  x  —  Sy^  gives  +  48a:y*,  or,  changing 
the  sign,  —  48icy*,  which  reduced  with  +  68ic?/*,  gives  20^?/*  for- 
a  remainder.  Again,  ~  80:^  X  —  ^y^  gives  + ,  and  changing  the 
sign,  —  64x^y^,  which  reduced  with  26x'^y^,  gives  —  S9x^y^, 
Hence,  the  result  of  the  first  operation  is  20xy'^  —  S9x^y^,  fol 
lowed  by  those  terms  of  the  dividend  which  have  not  been 
reduced  with  the  products  already  obtained.  For  the  second 
part  of  the  operation,  it  is  only  necessary  to  bring  down  the 
next  term  of  the  dividend,  to  separate  this  new  dividend  from 
the  primitive  by  a  line,  and  to  operate  upon  this  new  dividend  in 
the  same  manner  as  we  operated  upon  the  primitive,  and  so  on, 

THIRD   EXAMPLE. 

Divide  -  -  -  95a  -  TSa^  +  56c>  -  25  -  59a3  by  -3a' 
+  5  -  11a  +  7a3. 

56a4  _  59^3  _  73^2  +  95a  _  25 1  7a3  -  3a2  -  11a  +  5 


1st  rem.       -  35a3  +  15a2  +  55a  -  25 


8a 


2d   remainder      -      -     0. 


GENERAL   EXAMPLES. 

1.  Divide     lOaft  +  15ac  by  5a.  Ans.  2b  +  3<?. 

2    Divide    30aa?  —  54a;  by   6x,  Ans.  5a  —  9. 

3.  Divide     lOx^y  —  I5y^  —  5y  by   5y.  Arts.  2x^  —  3?/  ~  1. 

4.  Divide     12a  +  3aa;  —  ISaa;^  by   3a.  Ans.  4  +  a:  —  (yx^ 


18  ELEMENTS   OF  ALGEBRA.  LCHAP.   II 

m 

5.  Divide     6ax^  +  9a^x  +  ct^^^  by   ax,    Ans,  6a;  -f  9a  +  ax» 

6.  Divide    a^  +  2«^  +  x^  hj   a  +  x,  Ans.  a  +  x. 
1,  Divide     a^  —  Sa^y  +  3a?/2  —  y3  ^y   a  —  y. 

^Tis.  a^  —  2ay  +  y^« 

8.  Divide     24ca"b  —  12a3c62  —  6a&   by    ~  6ab. 

J        ^n5.   —  4a  +  2a2c6  +  1. 

9.  Divide  6x^  —  96  by  Sx  —  6.    ^W5.  2a;3  +  4x^  +  8a;  +  16. 

10.  Divide       -       -       a^  —  5a%  +  lOa^oJ^— lOa^a;^  +  5aa;'^  —  aj« 
by    a^  —  2aa;  +  x^,  Ans.  a^  —  Sa^x  +  3aa;2  _  ^3^ 

11.  Divide     48a;3  —  76aa;2  —  64a2a;  +  lOSa^   by   2a;  —  8a. 

Ans.  24a;2  —  2aa;  —  SSa^. 

12.  Divide   y^  —  Sy^a;^  +  Sy^x^  —  x^  by  y^  —  Sy^x  +  Bya;^  —  o?^- 

Ans.  y^  +  Sy^o;  +  Syx^  +  or^. 

13.  Divide     QAa'^b^ -25a^^  by   Sa^J^  +  5a6*. 

^^5.  Sa^js  —  5a6*. 

14.  Divide     6a3  +  23a2^  +  22a62  4- 6^3   by   3a2  +  4a6  +  ^>^. 

Ans.  2a  +  5b, 

1 5.  Divide     6aaj6  +  Qax'^y^  +  42a2a;^   by   ax  +  5aa;. 

^715.  o;^  +  ^y^  +  '''aa;. 

16.  Divide    -15a^  +  S7a^d-29a^cf-20b'^d^-\-4Abcdf-Scy^ 
by   3a2  —  bbd  +  c/.  ^ns.   —  5a2  4-  4Jc?  —  8c/. 

17.  Divide     a;*  +  o^^y^  +  y^   by   a;^  —  a:y  +  y^. 

Ans.  x^  -\-  xy  -{-  y^, 

18.  Divide     a;*  —  y*  by   a;  —  y.      -4^15.  x^  +  o^^y  +  o^y^  +  2/^« 

19.  Divide     3a4  -  ^aW  +  Za^c^  +  55*  -  3^>2c2   by   a^  -  b\ 

Ans.  3a2  -  562  4.  3^2. 

20.  Divide     ^rx^  —  bx^y^  —  6a; V  +  ^^^3/^  +  15a;3y3  —  9a; V 
"f  10a;2y5  4-  l^y^   by   3a;3  +  2a;2y2  +  3y2. 

Ans.  2x^  —  3a;2y2  _|^  5^3 


CHAP.   II.l  DIVISION.  49 

REMARKS    ON   THE   DIVISION   OF   POLYNOMIALS. 

68»  The  exact  division  of  one  polynomial  by  another  is  impossible: 

1st,  When  the  first  term  of  the  arranged  dividend  or  the  first 
term  of  any  of  the  remainders,  is  not  exactly  divisible  by  the  first 
term  of  the  arranged  divisor. 

It  may  be  added  with  respect  to  polynomials  that  we  aiiu 
often  discover  by  mere  inspection  that  they  are  not  divisible. 
When  the  polynomials  contain  two  or  more  letters,  observe  the 
two  terms  of  the  dividend  and  divisor,  which  contain  the  highest 
powers  of  each  of  the  letters.  If  these  terms  do  not  give  an 
exact  quotient,  we  may  conclude  that  the  exact  division  is  iia 
possible. 

Take,  for   example, 

12a3  -  5«26  +  7ab^  -  IW  \\4.a'^  +  Sab  +  Sb\ 

Bj  considering  only  the  letter  a,  the  division  would  appear 
possible;  but  regarding  the  letter  6,  the  exact  division  is  impos- 
sible, since  —-  ll^^  is   not   divisible  by  36^. 

2d.  When  the  divisor  contains  a  letter  which  is  not  in  the  dividend. 

For,  it  is  impossible  that  a  third  quantity,  multiplied  by 
one  which  contains  a  certain  letter,  should  give  a  product  inde- 
pendent of  that  letter. 

3c?,  A  monomial  is  never  divisible  by  a  polynomial, 

F(ir,  every  polynomial  multiplied  by  either  a  monomial  or  a 
polynomial  gives  a  product  containing  at  least  two  terms  whidi 
are  not   susceptible   of  reduction. 

4:ih,  If  the  letter,  with  reference  to  which  the  dividend  is  ar- 
ranged, is  not  found  in  the  divisor,  the  divisor  is  said  to  be  inde^ 
pendent  of  that  letter ;  and  in  that  case,  the  exact  division  is 
impossible,  unless  the  divisor  will  divide  separately  the  co-efficients 
of  the  different  powers  of  the  leading  letter. 

For  example,  if  the  dividend  were 

36a4  +  96a2  +  125, 
arranged  with  reference   to   the  letter  a,  and  the  divisor  Zb,  the 
divisor  would  be  independent  of  the  letter  a;  and  it  is  evident 

4 


50  ELEMENTS   OF  ALGEBRA.  [CHAP.   IL 

• 

that  the  exact  division  could  not  be  performed  unless  the  co- 
efficients of  the  different  powers  of  a  were  exactly  divisible  by  3i. 
The  exponents  of  the  different  powers  of  the  leading  letter 
in  the  quotient  would  then  be  the  same  as  in  the  dividend. 

EXAMPLES. 

i.  Divide    \^a^x^ '-2,^d?x^  —  VZax    by    6a:. 

%  Ans.  Za?x  —•  ^a^x^  —  2a, 

2.  Divide    25a*6  -  SOa^S  +  40a6    by    55. 

Ans.  5a*  — 6a2  +  8a. 

From  the  3d  remark  of  Art.  46,  it  appears  that  the  teim  of 
the  dividend  containing  the  highest  power  of  the  leading  letter 
and  the  term  containing  the  lowest  power  of  the  sam«>  letter 
are  both  derived,  without  reduction,  from  the  multiplication  of  a 
term  of  the  divisor  by  a  term  of  the  quotient.  Therefore,  nothing 
prevents  our  commencing  the  operation  at  the  right  instead  of 
the  left,  since  it  might  be  performed  upon  the  terms  containing 
the  lowest  power  of  the  letter,  with  reference  to  which  the  ar- 
rangement has   been   made. 

Lastly,  so  independent  are  the  partial  operations  required  by 
the  process,  that  afber  having  subtracted  the  product  of  the  divi- 
sor by  the  first  term  found  in  the  quotient,  we  could  obtain 
another  term  of  the  quotient  by  arranging  the  remainder  with 
reference   to   some   other  letter  and  then  proceeding  as  before. 

If  the  same  letter  is  preserved,  it  is  only  because  there  is  no 
reason  for  changing  it ;  and  because  the  polynomials  are  already 
arranged  with  reference  to  it. 

OF    FACTORING    POLYNOMIALS. 

69f  When  a  polynomial  is  the  product  of  two  or  more  factors, 
it  is  often  desirable  to  resolve  it  into  its  component  factors. 
This  may  often  be  done  by  inspection  and  hj  the  aid  of  the 
formulas  of  Art.  47. 

When  one  factor  is  a  monomial,  the  resolution  may  be  effected 
by  writing  the  monomial  for  one  factor,  and  the  quotient  arising 


THAP.   II.]  DIVISION.  51 

from  the  division  of  the  given  polynomial  t)j  this  fajjtor  for  the 
other  factor. 

1.  Take,  for  example,  the  polynomial 

ab  +  ac, 
In  which,  it  is  plain,  that  a  is  a  factor  of  both  terms :  hence 
ab  -}-  ac  =  a  (b  +  c). 

2.  Take,  for  a  second  example,  the  polynomial 

ab^c  +  6ab^  +  aiV. 

It  is  plain  that  a  and  5^  are  factors  of  all  the  terms :   hence 

ab^c  +  5a53  ^  ^52^2  _  ^52  (^  +  56  +  c^), 

3.  Take  the  polynomial  26a^  —  SOa^  +  l^a^b^ ;  it  is  evident 
that  5  and  a^  are  factors  of  each  of  the  terms.  We  may,  there- 
fore, put  the  polynomial  under  the  form 

5a2  (5a2  -  6ab  +  3^2), 

4.  Find  the   factors   of  Sa^b  +  9a^c  +  18a2a;y. 

Ans.  3a2  (^  +  3c  +  6iry) 

5.  Find  the  factors   of  Sa^cx  —  ISacx'^  +  2ac5y  —  SOa^c^^. 

Ans.  2ac  {4ax  —  9a;2  +  c*y  —  15a^c^a;). 

6.  Find  the   factors   of  24:a'^b^cx  —  S0a%^c^7/  +  SQa^^cd  +  6abc. 

Ans.  6abc  {4abx  —  baPb^c^y  +  Qa%H  +  1). 

By  the  aid  of  the  formulas  of  Art.  48,  polynomials  having 
certain  forms  may  be  resolved  into  their  binomial  factors. 

1.  Find  the  factors  of   o?-  +  2ab  +  62. 

Ans.  {a+  b)  X  {a  +  b) 

2.  49a;*  +  56a:3y  +  IQx^y^         =  {7x^  +  4.xy)  (7x^  +  4xy). 

3.  Find  the  factors  of  a^  —  2ab  +  b^. 

Ans.  (a  —  6)  X  (a  —  b). 

4.  64a262c2  -  48a6c2(^  +  9c^d^  =  {Sabc  -  Zcd^)  {Sabc  -  Scd^). 

5.  Find  the  factors  of  a^  —  b\  Ans.  {a  +  b)  X  {a -^  b). 
0    16a2c2-9d'4                           =  (4(ic -L  3c^2)  (4ac  ~  3^2). 


62  ELEMENTS  OF  ALGEBRA-  [CHAP.   IL 

GENERAL    EXAMPLES. 

1.  Find    the  factors  of  the  polynomial    6a^  +  Sa^h^  —  16a6' 

2.  Find  the  factors  of  the  polynomial    15a6c2  —  2hc^  +  Oa^i^i^ 
-^  I2db^c\ 

8.  Find    the    factors    of    the    polynomial    25a^Z>c^  —  ^Oa^c'^d 
-  5ac^  —  60ac6. 

4.  Find  the  factors  of  the  polynomial  4:2a'^P  —  labcd  +  lahd 

Arts,  lab  {Q^ab  —  cd  +  d), 

5.  Find  the  factors  of  the  polynomial  n^  +  2/i^  +  n. 
First,  n^  +  2n^  +  n  =  n  {n^  +  27^  +  1) 

=  n{n   +1)   X  (^  +  1) 
z=n{n   +  1)2. 

6.  Find  the  factors  of  the  polynomial  6a^bc  +  lOab^c  +  15abc\ 

Arts,  5abc  (a  +  26  +  3c). 

7.  Find  the  factors  of  the  polynomial  a^x  —  x^. 

Ans,  X  {a  -\-  x)  {a  —  xj. 

60.  Among  the  different  principles  of  algebraic  division,  there 
is  one  remarkable  for  its  applications.     It  is  enunciated  thus : 

The  difference  of  the  same  powers  of  any  two  quantities  is  ^ooactly 
divisible  by  the  difference  of  the  quantities. 

Let  the  quantities  be  represented  by   a  and  b ;   and  let  m  de 
note   any  positive  whole  number.     Then, 

a^  »_  j« 

will*  express  the  difference  between  the  same  powers  of  a  acid  A, 
and  it  is  to  be  proved  that  a^  — -  b"^  is  exactly  divisible  b)    a  —  & 
If  we  begin  the    iivision  of 

a*"  —  6*"  by  a  —  6, 
\\e  have 


ftWl    __    JTO  I 


a-^b 


1st  rem. a^**^'^b  —  b^ 

or,  by  factoring    -    -    -      b(aP-~^    —  6"^^). 


CHAP.  II.]  DIVISION.  53 

Dividing   a^   by   a   the  quotient  is  a*»~^,   by   the   rule  foi    the 

exponents.     The  product  of  a  —  5  by  aJ^~^  being  subtracted  from 

the  dividend,  the  first  remainder  is  a^'^h  —  6"*,  which  can  be 
put  under   the  form, 

Now,  if  the   factor 

of  the  remainder,  be  divisible  by  a  —  b,  b  times  (a^*~'^  —  ^"^'Oj 
must  be  divisible  by  a  —  b,  and  consequently  a^  —  b^  nmst 
also  be  divisible  by   a  —  b.     Hence, 

If  the  difference  of  the  same  powers  of  two  quantities  is  exactly 
divisible  by  the  difference  of  the  quantities^  then,  the  difference  of 
the  powers  of  a  degree  greater  by  1  is  also  divisible  by  it. 

But  by  the  rules  fbr  division,  we  know  that  a^  —  5^  is  divis 
ible  by  a  —  5 ;  hence,  from  what  has  just  been  proved,  a^  —  P 
must  be  divisible  by  a  —  b,  and  from  this  result  we  conclude 
that  a*  —  b^  is  divisible  hj  a  —  b  and  so  on  indefinitely :  hence 
the  proposition  is  proved. 

61.  To  determine  the  form  of  the  quotient.  If  we  continue 
the  operation  for  division,  we  shall  find  a'^~^  for  the  second 
term  of  the  quotient,  and  a^~^^  —  b^  for  the  second  remainder ; 
also,  a^~W  for  the  third  term  of  the  quotient,  and  a^~^b^  —  b^ 
for  the  third  remainder;  and  so  on  to  the  m**  term  cf  the  quo 
tient,  which  will  be 

and    the  m*^   remainder  will  be 

f^m-m^m  _  Jm     qj.      Jm  —  Jot  -__  Q^ 

Since  the  operation  ceases  when  the  remainder  becomes  0,  we 
sha_l  have  m  terms  in  the  quotient,  and  the  result  may  be  writr 
ten  thus : 

^"^""T  =  a*^^  +  a"^^^  +  a'""^^^  + +  dh"^   +  ^*"^- 

a  —  6 


CHAPTER  m. 

OF     ALGEBRAIC     FRACTIONS. 

62 •  An  algebraic  fraction  is  ai  expression  of  one  or  more 
equal  parts  of  1. 

One  of  these  equal  parts  is   called  the  fractional  unit.     Thus, 

■7^  is  an  algebraic  fraction,  and  expresses  that  1  has  been  divided 

into   h   equal  parts   and  that  a  such  parts  are  taken. 

The  quantity  a,  written  above  the  line,  is  called  the  numer- 
ator;  the  quantity  6,  written  below  the  line,  the  denominator ; 
and  both  are  called  terms  of  the  fraction. 

One  of  the  equal  parts,  as  —,    is    called   the  fractional  unit; 

and   generally,    the   reciprocal   of   the    denominator   is   the   frac- 
tional  unit. 

The  numerator  always  expresses  the  number  of  times  that  the 
fractional  unit  is  taken ;    for  example,  in   the  given  fraction,  the 

fractional  unit  -7-  is  taken  a  times. 
0 

63 •  An  entire  quantity  is  one  which  does  not  contain  any 
fractional  terms ;   thus, 

a^h  +  ex  is  an   entire   quantity. 

A  mixed  quantity  is  one  which  contains  both  entire  and  fraa 
tional  terms ;   thus, 

a^h  -|-  —   is  a  mixed  quantity. 

Every  entire  quantity  can  be  reduced  to  a  fractional  form 
having  a  given  fractional  unit,  by  multiplying  it  by  the  denomi- 
nator of  the  fractional  unit  and  then  writing  the  product  over  the 
denominator ;  thus,  the  quantity  c  may  be  reduced  to  a  fractional 


CHAP.   III.]  ALaEBEAIC  FRACTIONS.  55 

form  with  the  fractional  unit    -7-,   by   multiplying       by   b  and 

be 

dividijig  the  product  by  6,   which  gives  — . 

64i  If  the  numerator  is  exactly  divisible  by  the  denominator, 
a  fractional  expression  may  be  reduced  to  an  entire  one,  by  sim- 
ply performing  the  division  indicated;  if  the  numerator  is  not 
exactly  divisible,  the  application  of  the  rule  for  division  will 
sometimes  reduce   the  fractional  to   a  mixed  quantity. 

65.  If  the  numerator  a  of  the  fraction  —  be  multiplied  by 
any  quantity,  q,  the  resulting  fraction  -~  will  express  q  tim^a 
as  many  fractional  units  as  are  expressed  by  —  ;    hence: 

Multiplying  the  numerator  of  a  fraction  by  any  quantity  is 
equivalent  to  multiplying  the  fraction  by  the  same  quantity. 

66.  If  the  denominator  be  multiplied  by  any  quantity,  g,  the 
value  of  the  fractional  unit,  will  be  diminished   q  times,  and  the 

resulting  fraction    —    will   express  a  quantity  q  times  less  than 

the  given  fraction ;  hence : 

Multiplying  the  denominator  of  a  fraction  by  any  quanUty,  is 
equivalent  to  dividing  the  fraction  by  the  same  quantity, 

67 •  Since  we  may  multiply  and  divide  an  expression  by  the 
same  quantity  without  altering  its  value,  it  follows  from  Arts, 
65  and  QQ,  that : 

Both  numerator  and  denominator  of  a  fraction  may  be  multiplied 
by  the  same  quantity^  without  changing  the  value  of  the  fraction. 

In  like  manner  it  is   evident  that: 

Both  numerate r  and  denominator  of  a  fraction  may  be  divided 
by  the  same  quantity  without  changing  ihe  value  of  the  fraction. 

68.  We  shall  now  apply  these  principles  in  deducing  rules 
for  the  transformation  or  reduction  of  fractions. 


56  ELEMENTS  OF  ALGEBRA..  [CHAP.   IIL 

I.  A  fractional  is  said  to  be  in  its  simplest  form  when  the  numer- 
ator and  denominator  do  not  contain  a  common  factor.  Now, 
since  both  terms  of  a  fraction  may  be  di voided  by  the  .same 
quantity  without  altering  its  value,  we  have  for  the  reduction 
of  a  fraction  to  its  simplest  form  the  following 

RULE. 

Resolve  both  numerator  and  denominator  intc  their  simple  fao> 
tors  {Art,  59) ;  then^  suppress  all  the  factors  common  to  both 
terms,  and  the  fraction  will  be  in  its  simplest  form. 

Remark. — When  the  terms  of  the  fraction  cannot  be  resolved 
into  their  simple  factors  by  the  aid  of  the  rules  already  given, 
resort  must  be  had  to  the  method  of  thft  greatest  common  divi 
fe;or,  yet  to  be  explained. 

EXAMPLES. 

1.  Reduce  the  fraction     ^    ,  .   ..^     tc   H^  simplest  form. 
Sad  +  12a  ^ 

We  see,  by  inspection,  that  3  and  a  i»'A  f^otojt^?  of  the*  nu. 
merator,  hence, 

Sab  +  Qac  =  Sa{b  +  2e) 

We  also   see,  that  3   and  a   are  factors    vf  the  c\e'ioniina^^« 

hence, 

,      Sad+12a  =  Sa{d-\-4:), 

Sab  +  6ac  _Sa{b  +  2c)  _  b  +  2r 

^^^^'  Sad  +  12a  ~  Sa  (d  +  4)   ~  ^1 " 


2.  Reduce       r-^ — —7; — ;    to  its  simplest  form. 
9ab  +  Sad  ^ 


Ans, 


255c  +  56/" 
S.  Reduce       ^^^^  -,    ■..:,     to  its  simplest  form. 
Soo^  -f-  loo 


2ah  f  c 
2^    -f-  J 


Ans    ^±J-. 
Ans,  ^^  —  - 

o4floc 
4,  Reduce  — -j    to  its  simplest  form. 

Ans.  /, — |— 


CHAP.  III.] 


ALGEBRAIC  FRACTIONS. 


57 


5.  Reduce       Trr-rn — ^   ^^  i^s  simplest  form. 

84ao2 


Am. 


3a +/ 


0.  Reduce       ^^   ,^  . — r-;r-r    'O  its  simplest  form. 


12cc^/  +  4.CH 


Arts, 


7.  Reduce      -^^ — 77—=^     io  its  simplest  form. 


27ac2  —  6ac3 


u4^s. 


3a  —  J 

6ac  —  / 
9c  -  2c2*    , 


II.  From  what  was  shown  in  Art.  63,  it  follows  that  we  may 
reduce  the  entire  part  of  a  mixed  quantity  to  a  fractional  form 
with  the  same  fractional  unit  as  the  fractional  part,  by  multiply- 
ing and  dividing  it  by  the  denominator  of  the  fractional  part. 
The  two  parts  having  then  the  same  fractional  unit,  may  be 
reduced  by  adding  their  numerators  and  writing  the  sum  obtained 
over  the  common  denominator. 

Hence,  to  reduce  a  mixed  quantity  to  a  fractional  form,  we 
have  the 

RULE. 

Multiply  the  entire  part  by  the  denominator  of  the  fraction: 
then  add  the  product  to  the  numerator  and  write  the  sum  over  the 
denominator  of  the  fractional  part. 


Here, 


EXAMPLES. 


1.  Reduce      x  ~  -^^ ^ 


to  the  form  of  a  fraction. 


g^  —  x^  _x^  —  (a2  __  a;2)   ^  2a;2  —  a» 


%  Reduce      ar- to  the  form  of  a  fraction. 


2a 


Ans, 


ax  —  X* 


58 


3.  Reduce 


I.  Reduce 


ELEMENTS  OF  ALGEBEA. 

2a;-  7 


[CHAP    III 


Sx 


to  the  form  of  a  fraction. 


A71S, 


17a;  -  7 
Sx     • 


5.  Reduce       1  +  2a;  • 


6.  Reduce      Sx  —  I 


a;  —  3 

6x 


to  the  form  of  a  fraction, 

2a  —  a;  -f  1 

Ans.  . 

a 

to  the  form  of  a  fraction. 

10a;2  +  4a;  +  3 


-4ns. 


5a; 


X  +  a 
3a— 2 


to  the  form  of  a  fraction. 


^W5. 


9aa;  —  4a  —  7a;  +  2 
3a  -  2 


Remark. — We  shall  hereafter  treat  mixed  quantities  as  though 
thej  were  fractional,  supposing  them  to  have  been  reduced  to  a 
fractional  form  by  the  preceding  rule. 

III. — From  Art.  64,  we  deduce  the  following  rule  for  reducing 
a  fractional  to  an  entire  or  mixed  quantity. 

RULE. 

Divide  the  numerator  by  the  denominator^  and  continue  the  oper 
ation  so  long  as  the  first  term  of  the  remainder  is  divisible  by  the 
first  term  of  the  divisor :  then  the  entire  part  of  the  quotient  found  ^ 
added  to  the  quotient  of  the  remainder  by  the  divisor^  will  be  the 
mixed  quantity  required. 

If  the  remainder  is  0,  the  division  is  exact,  and  the  quotient 
is  an  entire  quantity,  equivalent  to  the  given  fractional  expres- 
sion. 

EXAMPLES. 

1.  Reduce    to  a  mixed  quantity. 


Ans,  =r  a  H . 

% 


CHAP,   III.J 

ALGEBRAIC    FKACTIONS.                                   { 

2.  Eeduce 

ax  —  x^                    ^.                 .J 

to  an  entire  or  mixed  quantity. 

-472.S.  a  —  a?. 

3.  Reduce 

to  a  mixed  quantity. 

Ans»  a =-• 

0 

4.  Reduce 

aP"  —  x"^ 

to  an   entire  quantity. 

Ans,  a  +  X. 

59 


5.  Reduce    —    to  an   entire  quantity. 

X      y 

Ans,  x"^  +  xy  +  y^. 

6.  Reduce to  a  mixed  quantity. 

DX 

3 
Ans,  2x  —  \  +  -r-. 

f  DX 

IV.  To  reduce  fractions  having  different  denominators  to  equiv 
alent  fractions  having  a  common   denominator. 

Let  -T-,  —   and  —,   be  any  three  fractions  whatever. 
^      ^  J 

It  is  evident  that  both  terms  of  the  first  fraction  may  be  mul 

nrl/ 

tiplied  by   df  giving  7-^,    and    that    this    operation    does    not 

change  the  value  of   the  fraction   (Art.   67). 

In    like  manner  both  terms  of   the    second  fraction  may  be 

hcf 
multiplied  by   hf^   giving  j^  ;   also,  both   terms  of  the  fraction 

odj 

--r  may  be  multiplied  by   hd^  giving  -z-^. 

If   now  we    examine    the    three  fractions   r^^,     7—    and    rrrA 

hdf      hdf  hdf 

we  see  that  they  have  a  common  denominator,  hdf^  and  that 
each  numerator  has  been  obtained  by  multiplying  the  numerator 
of  the  corresptnding  fraction  by  the  product  of  all  the  denom- 
inators except  its  own.  Since  we  may  reason  in  a  similar 
manner  upon  any   fractions   whatever,   we  have  the  following 


60  ELEKENTS  OP  ALGEBRA.  [CHAP.   III. 

BULK 

Multiply  each  numerator  into  the  product  of  all  the  denomina^ 
tors  except  its  own^  for  new  numerators^  and  all  the  denominator  $ 
together  for  a  common  denominator. 

EXAMPLES. 

1.  Eeduce    -7-    and    —   to  equivalent  fractions  having  a  com 

0  c 

mon  denominator. 

a  X  c  =ac)     ^ 

,0  r  the  new  numerators. 
b  X  b  =  b^  ) 

and    -     b  X  c  z=  be      the  common  denominator. 

2.  Reduce  -7-  and  to  equivalent  fractions  having  ^  com 

b  c  ^ 

.        ac         ^    ab  +  b^ 
mon   denommator.  Ans,  r—  and 


be  be 

on/l       /7       i-r\      nmiiTrQlnnf. 

2a'    3c 


3.  Reduce    — ,   —   and'^  d,   to    equivalent  fractions   having   a 


.         9cx      Aab        T  Q>acd 
common  denommator.  Ans.  ~ — ,   -- —  and  — — . 

bac      bac  bac 

3  2.1J  2a; 

4.  Reduce    -7-,  -tt    and    a  H ,    to  equivalent  fractions  hav- 

4  3  a 

9a      Sax        ,    I2a^ -\- 24:X 

mff  a  common  denommator.     A71S.  777-,   -ttt-   ana    — ; . 

°  12a'    12a  12a 

1  a^  a   ~\~  X 

5.  Reduce    — ,  -rr-   and    ,   to   equivalent  fractions  hav- 

2  3  a  +  X 

ing  a  common  denominator. 

3a  +  3a;    2a3  +  2a^^  Ga^  J-  6x^ 

^'''-  6M=^'      Qa  +  6x      ^"""^    Qa  +  6x' 

6.  Reduce    ; ,    and    — ,   to  equivalent  fractions  hav 

a  —  b       ax  c 

mg  a   common  denominator. 

a^cx  ac^  —  abc  —  bc'^  +  cb'^        ^  a%x  ~  ab'^x 

Ans.    —z r~?    V 1 ^^^    — 5 1 — "• 

a^cx  —  abcx  a^cx  —  abcx  aHx  -   abcx 


CHAP.   III.J  ALGEBRAIC    FRACTIti^S.  61 

V.  To   add  fractions  together. 

Quantities  cannot  be  added  together  unless  they  have  the 
same  unit.  Hence,  the  fractions  must  first  be  reduced  to  equiv- 
alent ones  having  the  same  fractional  unit;  then  the  sum  of 
the  numerators  will  designate  the  number  of  times  this  unit 
is  to  be  taken.  We  have,  therefore,  for  the  addition  of  frac. 
tions  the  following  , 

RULE. 

Meduce  the  fractions,  if  necessary,  to  a  common  denominator : 
then  add  the  numerators  together  and  place  their  sum  over  the 
tommon   denominator, 

EXAMPLES. 

1.  Find  the  sum  of  -r-,   -7-    and    — . 
0      d  f 

Here,    -     a  X  d  xf  =  adf^ 

c  X  h  xf  =  cbf  >    the  new  numerators. 

e  Xh  X  dzzzebd) 
And     -     b  X  d  xf  =  bdf      the  common  denominator. 
adf      cbf  ,    ebd      adf  +  cbf  +  ebd    , 

^^"^^'  bif-^Wf-^bdf^-^—bk —  ^'^  ^""• 

2.Toa-?^'    add    5  +  ?^.        Ans,a  +  b+^-^^^^:^. 
be  be 

3.  Add  — ,   — -    and    --    together.  Ans,  x  +  ■^, 

z      s  ^  12 

4.  Add  —^   and    y    together.  Ans.  -— ~. 

5.  Add.  +  ^-^    to    3.+  ?if^.    Ans.  4.  +  12l=iI. 

o  4  12 

6.  It  is  required  to  add    4x,   -^    and    ^  together. 

J         ^      .   5a;3  -{-  ax  +  a^ 

Ans,  4:X  -\ , 

2ax 

7.  It  is  required  to  add    — .  —    and      ^  7"       together. 

Ans.  2x  i ^ , 


52  ELEMENTS  OF  ALGEBRA.  [CHAP.   IIL 

8.  It  is  re<iuired  to  add   4x,   —    and  2  -i  '—-  together. 

-4715.  4a;  H . 

45 

9.  It  is  required  to  add   3ic  +  — -    and    x  — --    together. 

Ans,    Sx  +  -r=-. 
45 

10.  What  is  the  sum  of    -,,   — — r    and 


Ans, 


a  —  V    a  -\-  b  a  -{-  x' 

a^  —  ax^  +  a^b  —  bx"^  +  a^c  +  «ca;  —  abc  —  6ca;  +  «^c?  —  6^df 


^  a^  -]-  a^  {b  -\-  c  4-c?)  —  a  (a;2  —  co:  +  6c)  —  b  (x^  +  ex  -{-  bd) 

VI.  To  subtract  one  fraction  from  another. 

Reduce  the  fractional  quantities  to  equivalent  ones,  having  the 
same  fractional  unit ;  the  difference  of  their  numerators  will 
express  how  many  times  this  unit  is  taken  in  one  fraction  more 
than  in  the  other.     Hence  the  following 

RULE. 

I.  Reduce  the  fractions  to  a  common  denominator, 

II.  Subtract  the  numerator  of  the  subtrahend  from  the  numer- 
ator of  the  minuend^  and  place  the  difference  over  the  common 
denominator, 

EXAMPLES. 

•     ^  X  —  a        ^  ^      ^     2a  —Ax 

1.  From   -    -    -     — ^r^ —     subtract      — . 

Zo  6c 

(x  —    a)  X  3c  =  Zcx  —  3ac  )      , 
Here,      .\         .  \      _,       .  ,      or     r    ^'^^  numerators. 
'      (2a  —  Ax)  X  26  =  Aab  —  ^bx  ) 

And,  2b   X  3c  =  Qbc        the  common  denominates 

3ca;  —  3ac        Aab  —  ^bx   ___  3ca;  —  3ac  -—  4a6  +  86a: 
^^''^®'  Wc  Wc         ""  Wc  • 

2.  From    -    -     -=--  subtract   -^.  Ans.  -^rr-. 

7  9  •H5 


CHAP.   III.] 

ALGEBRAIC  FRACTIONS. 

( 

3.   From   - 

.    5y 

subtract  -^.     .               Ans. 
o 

37y 
8  • 

4.   From   - 

Sx 

7 

subtract   — .                       A^is. 

13a! 
63' 

5.  From   - 

X  +  a 
b 

c            .        dx  +  ad 
subtract  --r*       -^^5-  rr 

-be 

6.   From   - 

Sx+  a 
'         56 

subtract  — ^ — . 
o 

24a;  +  8a  —  106a;  - 
'^'^-                   406         , 

-356 

7.   From   - 

-     3.  +  I 

c 

cx  +  bx 
Ans.   2x  H =— 

-a6 

63 


VII.  To  multiply  one  fractional  quantity  by  another. 

Qi  C 

Let   -  represent  any  fraction,  and  -   any  other  fraction;   and 
let  it  be  required  to  find  their  product.  ^ 

If,  in  the  first  place,  we  multiply  -  by   c,   the  product  will 

be  "Y",  obtained  by  multiplying  the  numerator  by  c,  (Art.  G5); 
6 

but    this    product    is   d   times   too    great,   since    we    multiplied 

-  by  a  quantity  d  times   too   great.     Hence,  to   obtain  the  true 

product  we   must   divide  by  d,   which  is   effected   (Art.  66)  by 
multiplying  the  denominator  by  d.     We  have  then. 

a        c       ac      . 

b^'d^M^   ^^^^^ 


RULE. 

.      1.     Cancel   all  factors    common    to    the   numerator   and   denymir 
nator. 

II.  Multiply  the  numerators  together  for  the  numerator  of  the 
product,  and  the  denominators  together  for  the  denominator  of  the 
product. 


64  ELEMENTS  OF  ALGEBRA.  [CHAP,  in. 


EXAMPLES. 


I.  Multiply    a'\ by   -7-. 

_.    ^  bx       a^  +  hx 

First,     -    -    -    -     a-] = ■ ; 


T-T  a^  +  bx  c         a^c  +  ^cx 

Hence,     -    .     x  -r  = 5 5 

a  a  ad        ^ 

2.  Required  the  product  of     —    and   — . 

2ic  Sx^ 

8.  Required  the  product  of     ---    and   — -. 

4.  Find  the  continued  product  of    — ,     and 

^  a         c 


5.  It  is  required  to  find  the  product  of     b -\ and    — . 

(t  X 

.        ab  -\-  bx 
Ans, 


Ans, 

9ax 
2b' 

Arts, 

Sx^ 
5a' 

Sac 

2b' 

Ans. 

9ax 

x 


-r.       ...            -.          ^    x^-b'^        ^    a;2  +  62 
6.  Required  the  product  of    — 7 and    -j—r — • 


'^'''-  b\  +  bc^' 

X  ~\-  1  X  "~~  1 

7.  Required  the  product  of   a;  H and  ,. 

.        ax'^  —  ax  -{-  x^  —  1 

Ans.  r-, — 7 • 

a^  +  ab 

ax  ■^    ^^  —  ^^ 

8.  Required  the  product  of   a  H — — -    and    — jr~2' 

a^  {a  +  x) 


€HAP.  III.3  ALGEBRAIC  FRACTIONS.  65 

VIII.  To  divide  one  fraction  by  another. 

Let  —  represent  the  first,  and  —  the  second  fraction;   then 
0  d 

he  division  may  be  indicated  thus. 

(i) 
ii) 

If  now  we  multiply  both  numerator  and  denominator  of  tliis 
complex  fraction  by  — ,    which  will  not  change  the  value  of  the 

fraction  (Art.  67),  the  new  numerator  will  be  7-,  and  the  new 

be 

denominator  — ,  which  is  equal  to  1. 

(-]    (-] 

,x  «  c  \h )         \hcj        ad 

Hence,      -      -      "t"  -r  -r  =  7-^    =  ~-  =  ^. 
0  d         /  c\  1  he 

u) 

This  last  result  we  see  might  have  been  obtained  by  invertmg 
the  terms  of  the  divisor  and  multiplying  the  dividend  by  t)ie 
resulting  fraction.  Hence,  for  the  division  of  fractions,  we  have 
the   following 

RULE. 

Invert  the  terms  of  the  divisor  and  multiply  the  dividend  hy  the 
resulting  fraction, 

EXAMPLES. 
h  -f 

1.  Divide    -     -      -      a  —  —    by    ~. 


a  — 


2c      '      g' 

h        2ac- 


2c  2c 


Hence,     a  ^  ±  ^  I- =^-^  x  ^t  =  ?^f£^. 
2c       g  2c       "^  f  2cf 

2.  Let    ---    be  divided  by    --5.  Ans.  --777-. 

5  ''     13  00 


66 

ELEMENTS  OF  ALGEBRA.                  [CHAP.   Ill 

3.  Let 

4^2 

—     be  divided   by    5x, 

.       4x 
Ans.  -. 

4.  Let 

— - —    be  divided  by    -^. 

Ans.   ^+  \ 
4tx 

5,  Let 

X                                                         X 

be  divided  by    -r-, 

x  —  1                          •'2 

Ans.  -. 

X  —  \ 

6.  Let 

5aj     -      ...,,,       2a 
-—    be  divided  by   -^. 

5bx 
Ans.   ^r-. 
2a 

7.  Lot 

■  ^   ,      be  divided  by    -— r» 

Ans.     ^  _    . 

8.  Let 

/^4  __  J4 

-r — — — TT-    be  divided  by 

a;2  —  26a;  +  b^                           '' 

x^  +  bx 
a;-6  • 

Ans.  X'-\ . 

X 

9.  Divide by r-.    Ans, 

1  —  a;        •^      1  —  a;2 

ax{\  +  x)  —  X  —  \ 

,„.  BivM,    i±i    b,     l±i. 

Ans.  -  (1  4  a). 

69.  If 

we  have  a  fraction  of  the  form 
a 

we  may 

observe  that 

-7  =  —  c,    also        7  =  —  c    and 
6                           —  6 

—  a 

-  =  c ;    that  is, 

The  sign  of  the  quotient  will  be  changed  by  changing  the  sign 
either  of  the  numerator  or  denominator^  but  will  not  be  affected  by 
changing  the  signs  of  both  the  terms. 

70.  We  will  add  two  propositions  on  the  subject  of  fractions. 

L  If  the  same  number  be  added  to  each  of  the  terms  of  a  pro'per 
fraction^  the  .fraction  resulting  from  these  additions  will  be  greater 
than  the  first ;  but  if  it  be  added  to  the  terms  of  an  impropet 
fraction^  the  resulting  fraction  will  be  less  than  the  first. 

Let  the  fraction  be  expressed  by    —. 

Let  m  represent  the  number  to  be  added  to  each  term :  \hQ» 

the  new  fraction  will  be,        rr—. — . 

0  +  m 


CBAJP    III.]  ALGEBRAIC  FRACTIONS.  67 

In  order  to  compare  the  two  fractions,  they  must  he  reduced 
to   the   same  denominator,  which  gives  for 

a        ab  +  am 


the  first  fraction, 

and  for  the  new  fractioii, 


b        6^  +  bm 
a-\-  m      ab  -\-bm 


b  -\-  m       6^  _|_  5y^* 

Now,  the  denominators   being  the  same,  that  fraction  will  bo 

the    greater   which    has    the   greater    numerator.      But  the  two 

numerators   have  a  common  part  a6,   and  the   part   bm  of  the 

second  is  greater  than  the  part  am  of  the   first,   when   6  >  a : 

hence 

ab  -\'  bm  ^  ab  -\-  am ; 
that  is,  when  the  fraction  is  proper,  the  second  fraction  is  greater 
than  the  first. 

If  the  given  fraction  is  improper,  that  is,  if  a  >  6,  it  is  plain 
that  the  numerator  of  the  second  fraction  will  be  less  than  that 
of  the  first,  since   bm  would  then  be  less  than   am. 

II.  If  the  same  number  be  subtracted  from  each  term  of  a  proper 
fraction^  the  value  of  the  fraction  will  be  diminished ;  but  if  it  be 
subtracted  from  the  terms  of  an  improper  fraction^  the  value  of  the 
fraction  will  be  increased. 

Let  the  fraction  be  expressed  by    —^    and  denote  the  number 

to  be  subtracted  by  m. 

Then,  -; will  denote  the  new  fi-action. 

b  —  m 

By  reducing  to  the   same   denominator,  we  have, 

a       ab  —  am. 


and 


h  -  62  -  bm ' 
-—  m      ab  —  bm 


b  ^  m       b"^  —  bm' 
Now,  if  we  suppose   a<ib,   then   am  <^  bm-,    and  if  am  <  bm^ 

then  will 

ab  —  am  "^  ab  —  bm: 

that  is,  the  new  fraction  will  be  less  than   the  first. 

If  a  >  6,   that  is,  if  the  fraction  is  improper,  then 

am  >  bm,    and    ab  —  am  <C.ab  —-  bm, 

that  is,  the  new  fraction  will  be  greater  than  the  first. 


68  "  ELEMENTS  OF  ALGEBRA*.  LCHAP.  HI, 


GENERAL    EXAMPLES. 

1    Add 5    to    — - — -,  Arts,   -Aj r^. 

11  2 

2.  Add    r-i —    to .  Ans, 


\  -{-  X  1  —  ic'  '  1  —  a;2' 

^^            a  +  6         ,a— 5  .  4a5 

3.  From    ;-     take    — --r«  Ans. 


a  —  b  a  +  b'  '    a?  ^b'^' 

4.  From take    r— : — ::.  Ans, 


r    Ttr  1  .  -,       a;2  _  9^  +  20   ^      0^2  -  13a;  +  42 
^-  ^^l^^Pl^         :.2,6a:        ^^  ^^  -  5a:       ' 

a;2-lla;  +  28 

-4n5. . 

x^ 

^    ■»«-,.  1  a;*  —  5*         ,       a;2  4-  ^^  ^  ,   .    to 

6.  Multiply    ^3+-2J^+65  ^7    -^rT'  ^'**-  ''+**• 

^_.  .T      a  +  ic.a-— a;,       a  +  x       a  —  x 

7.  Divide 1 ; —    by ; — . 

a  —  x       a  +  x     ''     a -J  X       a  +  x 

.        a^  +  x^ 

Ans,  —7; . 

2aa; 

8.  Divide    1  -\ by    1 — -.  Ans,  n, 

n  +  1       ''  n  +  1 


EXAMPLES   INDICATING   USEFUL   FORMS    OF   REDUCTION. 

-     _a_       £_       J_  ___  adfx^       chfx^      ebdx^ 

'    bx'^dx^'^fa^  ^bdfa^'^bdfx^^bdfx^ 

adfx^  +  bcfx    +  bde 

"  bdp 

2    _^  .    £ _f ^  _  adfhx^       bcfhofi        bedhx''       bdfg^ 

bx^  dx^     /«3      h^  ""  bdfhx^^  "^  bdfkx^"^  ■""  bdfk^'' ""  bdfhx^^ 
_  adfJix^  +  bcfhx^  —  bedhx    —  bdfg 


ICHAP.   III.  EXAMPLES  IN   FRACTIONS.  69 

i_+^     i-x^      __      (1  +  x^y  (1  -  x^Y 

^'  l~x^^  1  +  x^        '^  {l-x^){l  +  x^)     '^  (l^x^){l+x^) 
_  (1  +  x^Y  +  (1  -  (^^Y 
""  (1  -  x')  (1  +  x^) 

2(1  +  x^) 
~      1  -  a;4  • 


1                         1—x  I  -\-  X 

+    1 '       =  TT"; — wl 1\  + 


l  +  o;    ^      1-x      ■"  (l+a;)(l~a;)       {l+x){l-x) 

1  —  X  -]-  1  +  X 

-{l+x){l^x) 
2 


a  +  5  a-"6        _{a-{-bY-{a-bY 

\-b){c 
4ab 


o 

'  a  —  b         a  +  b  {a  +  b)  {a  —  b) 


4. 


l-hx^        l-x^     _        (1  +  a;^)^ (1  ~  a;2)2 

l-a;2         l+ic2     ""  (l-ir2)(l +a:2)       (1  -a;2)(l  +  a;2) 

_  (1  +  x'^Y  -  (1  -  ^^)^ 

""      (1  -  x^)  (1  +  a;2) 

4:X^ 


•  1  -  ic2    •    1  +  a;2     ■"  1  -  a;2   ^  1  -  a;2    ""  (1  ~  a;2)2- 
ic*  —  54  a;2  +  j^  a;*  —  6*  a;  —  5 


x^  —  2bx  t\-  b^  '      a;  —  6        a;2  —  26a;  +  ^2       a;2  _.  5a; 

(^4  _  54)  (^  _5) 


""  (a;2  —  2bx  +  b^)  (a;2  +  bx) 

_  (a;2  _  52)  (a;2  _|.  ^>2)  (^  __  5) 

~        (a;  —  6)2  a;  {x  +  b) 
_{x  +  b){x-  b)  (a;2  +  62)  (a;  ^  6) 
-        a;  (aj  -  6)  (a;  -  6)  (a;  +  6) 
x^  +  b^ 


70  ELEMENTS  OF  ALGEBEA.  LCHAP.  Uh 

Of  the  Symbols  0,    oo  and  — . 

71.  The  symbol    0  is   called   zero,  which  signifies  in  ordinary 

language,   nothing.      In   Algebra,   it   signifies   no  quantity :    it  is 

also   used   to  expres  a  quantity  less  than  any  assignable  quantity. 

The  symbol  oo  is  called  the  symbol  for  infinity ;  that  is,  it  is 

Ufecd  to  represent  a  quantity  greater  than  any  assignable  quantity. 

If  we   take  the  fraction  — ,  and   suppose,  whilst  the  value  of 

a  remains  the  same,  that  the  value  of  h  becomes  greater  and 
greater,  it  is  evident  that  the  value  of  the  fraction  will  become 
less  and  less.  When  the  value  of  b  becomes  very  great,  the 
value  of  the  fraction  becomes  very  small ;  and  finally,  when  b 
becomes  greater  than  any  assignable  quantity,  or  infinite,  the 
value  of  the  fraction  becomes  less  than  any  assignable  quantity, 
or  zero. 

Hence,  we   say,  that   a   finite   quantity   divided  by  infinity  is 
equal  to  zero. 

We  may  therefore  regard  — ,    and  0,    as  equivalent   symbols. 

If  in   the  same  fraction  —,  we  suppose,  whilst  the  value  of  a 

remains  the  same,  that  the  value  of  b  becomes  less  and  less,  it 
is  plain  that  the  value  of  the  fraction  becomes  greater  and 
greater;  and  finally,  when  b  becomes  less  than  any  assignable 
quantity,  or  zero,  the  ^alue  of  the  fraction  becomes  greater  than 
any  assignable  quantity,  or  infinite. 

Hence,  we  say,  that  a  finite  quantity  divided  by  zero  is  equal 
to  infinity. 

We  may  then  regard  --r-  and  oc  as  equivalent  symbols :  Zert 

and  infinity  are  reciprocals  of  each  other. 

The  expression  —  is  a  symbol  of  indetermination ;  that  is,  it 

is  employed  to  designat'3  a  quantity  which  admits  of  an  infinite 
number  of  values.  The  origin  of  the  symbol  will  be  explained 
in  the  next  chapter. 


OSAP.  III.J  ALGEBRAIC    FRACTIONS.  71 

It  should  be  observed,  however,  that  the  expression  —  is  not 

always  a  symbol  of  indetermination,  but  frequently  arises  from 
the  existence  of  a  common  factor^  in  both  terms  of  a  fraction, 
which  factor  becomes  zero,  in  consequence  of  a  particular  hypo- 
thesis. 

I.  Let  us  consider  the  value  of  x  in  the  expression 

a3  — 6 


X  : 


a2  ~  62^ 


If,  in  this  formula,  a  is  made  equal  to  b,  there  results 

0 

X  =  — , 
0 

But,     -     .     -     a^^b^=:[a'-b){a^  +  ab+  b^) 

and    -    -    a2  —  ^2  =  (a  —  5)  (a  +  5), 

hence,  we  have, 

_  (g  —  6)  (a2  +  ab+  b'^) 

''•"         {a-b){a  +  b)       • 

Now,  if  we  suppress  the  common  factor  a  —  b,  and  then  sup 
pose  a  =z  b,  we  shall  have 

3a 

2.  Let  us  suppose  that,  in  another  example,  we  have 
_  a2  _  52 
^  -  (a  -  6)2- 

If  we  suppose  a  =  b,   we  have 

0 

x==-^. 

If,  however,  we  suppress  the  factor  common  to  the  numerator 
ftnd  denominator,  in  the  value  of  x,   we  have, 
_{a-\-b){a^  b)       a  +  b 
'^  {a  —b){a  —  b)  "  a  —  b' 

Tf  now  we  make  a=:b,  the  value  of  x  becomes 

25 

-^  =  00. 


72  ELEMENTS   OF  ALGEBRA.  [CHAP.    IIT. 

3.  Let  us  suppose  in  another  example, 

{a  -  hy 

in  which  the  value  of  x  becomes  —  when  we  make  a  zz.b. 

If  we  strike  out  the  common  factor  a  —  b,  we  shall  find 

a  —  b 
^"^  o?^ab-{-  62* 

If  now  we  make  a  z=b,  the  value  of  x  becomes 

Therefore,  before   pronouncing  upon  the  nature  of  the  expres 

sion  —,  it  is  necessary  to   ascertain  whether  it  does  not  arise 

from  the  existence  of  a  common  factor  in  both  numerator  and 
denominator,  which  becomes  0  under  a  particular  hypothesis. 
If  it  does  not  arise  from  the  existence  of  such  a  factor,  we 
conclude  that  the  expression  is  indeterminate.  If  it  does  arise 
from  the  existence  of  such  a  factor,  strike  it  out,  and  then  make 
the  particular  supposition. 

If  A  and  JB  represent  finite  quantities,  the  resulting  value  of 
the  expression  will  assume  one  of  the  three  forms;  that  is: 
A  A  0 

¥'       -0    ''    a'^ 

it  will  be   either  finite^  infinite^  or  zero. 

This  remark  is  of  much  use  in  the  discussion  of  problems. 


CHAPTER  IV. 

EQUATICNS  OP  THE    FIRST   DEGREE    INYOLVING   BUT   ONE  UNKNOWN  QUANTITY 

72.  An  Equation  is  the  algebraic  expression  of  equality  bo- 
tween  two  quantities. 

Thus,  x=z  a-i-  b, 

is  an  equation,  and  expresses   that  the  quantity  denoted  by  x  is 
equal  to  the  sum  of  the  quantities  represented  by  a  and  b. 

Every  equation  is  composed  of  two  parts,  connected  by  the 
sign  of  equality.  The  part  on  the  left  of  this  sign  is  called  the 
first  member^  that  on  the  right  the  second  member.  The  second 
member  of  an  equation  is  often  0. 

73 •  An  equation  may  contain  one  unlcnown  quantity  only,  or 
it  may  contain  more  than  one.  Equations  are  also  classified 
according  to  their  degrees.  The  degrees  are  indicated  by  the 
exponents  of  the  unknown  quantities  which  enter  them. 

In  equations  involving  but  one  unknown  quantity^  the  degree  is 
denoted  by  the  exponent  of  the  highest  power  of  that  quantity  in 
any  term. 

In  equations  involving  more  than  one  unknown  quantity^  the 
degree  is  denoted  by  the  greatest  sum  of  the  exponents  of  the  unknown 
quantities  in  any  term, 

Eor  example: 
ax  -f- b  z=z  ex  +  d 
ax  +  Zby  +  cz+M^O 

aa;2  +  26ar  +  c  =  0 

oor^  +  bxy  +  cy^  +  c?  =  0 

c^x'^  +  2dgx^  =  abx  -—  c^ 
4aicy2  —  2c2/3  -|-  abxy  ==  3 
and  so  on. 


f    are  equations  of  the  first  degree. 
r    are  equations  of  the  second  degree. 
"    are   equations  of   the    third   degree, 


74  ELEMENTS  OF  ALGEBRA.  ICHAP.  IV 

• 

74.  Equations  are  likewise  distinguished  as  numerical  equations 
and  literal  equations.  The  first  are  those  which  contain  numbers 
only,  with  the  exception  of  the  unknown  quantity,  which  is 
always  denoted  by  a  letter.     Thus, 

4a;  —  3  =  2^  +  5,         ^x^--x  =  S, 
are  numerical  equations, 

A  literal  equation  is  one  in  which  a  part,  or  all  of  the  known 
quantities,  are  represented  by  letters.     Thus, 

hx^  +  ax  —  So:  =  5,       and      ex  +  dx^  =  c  +  /, 
are  literal  equations. 

75.  An  identical  equation  is  an  equation  in  which  one  member 
is  repeated  in  the  other,  or  in  which  one  member  is  the  result  of 
certain  operations  indicated  in  the  other.  In  either  case,  the 
equation  is  true  for  every  possible  value  of  the  unknown  quan- 
tities which  enter  it.      Thus, 

X^  —  y^ 

ttx  +  b  =  az+  6,         {x  +  cbY  =  x^+  2ax  -f  a^,        —  =  a;  —  y, 

are  identical  equations. 

76.  From  the  nature  of  an  equation,  we  perceive  that  it  must 
possess  the  three  following  properties : 

1st.  The  two  members  must  be  composed  of  quantities  of  the 
same  kind. 

2d.  The  two  members  must  be  equal  to  each  other. 

3d.  The  essential  sign  of  the  two  members  must  be  the  same. 

76 1"^  An  axiom  is  a  self-evident  proposition.  We  may  here 
enumerate  the  following,  which  are  employed  in  the  tra7isforma- 
tion  and  solution  of  equations : 

1.  If  equal  quantities  be  added  to  both  members  of  an  equation, 
the  equality  of  the  members  will  not  be  destroyed. 

2.  If  e^ual  quantities  be  subtracted  from  both  members  of  an 
equation,  the  equality  will  not  be  destroyed. 

3.  If  both  members  of  an  equation  be  multiplied  by  equal 
quantities,  the  products  will  be  equal. 

4.  If  both  members  of  an  equation  be  divided  by  equal  quan 
titles,  the  quotients  will  be  equal. 

5.  Like  powers  of  the  two  members  of  an  equation  are  equal 

6.  Like  roots  of  the  two   members  of  an  equation  are  equal. 


CHAP.  IV.]   EQUATIONS  OF  THE  FIKST  DEGREE.      ^    75 

'  Solution  of  Equations  of  the  First  Degree, 

77.  The  solution  of  an  equation  is  the  operation  of  finding  a 
value,  for  the  unknown  quantity  such,  that  when  substituted  for 
tne  unknown  quantity  in  the  equation,  it  will  satisfy  it ;  that  is, 
make  the  two  members  equal.  This  value  is  called  a  root  of 
the  equation. 

In  solving  an  equation,  we  make  use  of  certain  transformations, 

A  transformation  of  an  equation  is   an  operation  by  which  we 

Aange  its  form  without  destroying  the  equality  of  its  members. 

First  Transformation. 

78 •  The  object  of  the  first  transformation  is,  to  reduce  an 
equation^  some  of  whose  terms  are  fractional^  to  one  in  vjhich  all 
of  the  terms  shall  be  entire. 

Take  the  equation, 

2a;        3  a;        ,, 

First,  reduce  all  the  fractions  to  the  same  denominator,  by  the 
known  rule ;  the   equation  then  becomes 

72  ""  "72  "*"  "72  ~ 

if  now,  both  members  of  this  equation  be  multiplied  by  72, 
the  equality  of  the  members  will  be  preserved  (axiom  3),  and 
the  common  denominator  will  disappear ;     and  we  shall  have 

48a:  —  54rc  +  12a;  =  792 ;      or  by  dividing 
both  members  by  ^,       8a;  —    9a;  +    2a;  =  132. 

The  last  equation  could  have  been  found  in  another  manner 
by  employing  the  least  common  multiple  of  the  denominators. 

The  common  multiple  of  two  or  more  numbers  is  any  num- 
ber which  each  will  divide  without  a  remainder ;  and  the  least 
common   multiple,  is  the  least  number  which  can  be  so  divided. 

The  least  common  multiple  of  small  numbers  can  be  found 
by  inspection.  Thus,  24  is  the  least  common  multiple  of  4,  6 
and  8 ;  and   12  is  the  least  common  multiple  of  3,  4  and  6. 


76  ELEMENTS   OF  ALGEBRA.  [CHAP.   IV. 

• 

Take  the  last  equation, 

We  see  that  12  is  the  least  common  multiple  of  the  de- 
nominators, and  if  we  multiply  each  term  of  the  equation  by 
12,  reducing  at  the  same  time  to  entire  terms,  we  obtain 

8a;  —  9a;  +  2a;  =  132, 
the  same  equation  as  before  found. 

Hence,  to  transform  an  equation  involving  fractional  terms  to 
one  involving  only  entire  terms,  we  have  the  following 

RULE. 

JF^orm  the  least  common  multiple  of  all  the  denominators^  and 
then  multiply  both  members  of  the  equation  by  it,  reducing  fractional 
to  entire  terms. 

This  operation  is   called   clearing  of  fractions. 

EXAMPLES. 

1.  Eeduce    -^  +  -r 3  =  20,    to   an  equation  involving  only 

entire   terms. 

We  see,  at  once,  that  the  least  common  multiple  is  20,  by 

which  each  term  of  the   equation  is   to   be  multiplied. 

^       ^^  20       , 

Now,  —  x20  =  x  X  -^  =  4:X, 

and  —■  X  20  =  X  X  -r  =  5x: 

4  4  ' 

that  is,  we  reduce  the  fractional  to   entire   terms,  by  multiplying 

the   numerator  by  the  quotient  of  the  common  multiple  divided  by 

the  denominator,  and  omitting  the  denominators. 

Hence,  the  transformed   equation  is 

4a;  +  5a;  —  60  =  400. 

2.  Eeduce    -r-  +  =-  —  4  =  3     to  an   equation   involving  only 

O  7 

entire  terms.  Ans,  7a;  +  5a;  —  140  =  105. 


CHAP,   ly.J         EQUATIONS  OF  THE   FIRST  DEGREE.  77 

CL  C 

3.  Reduce    -7 —-{-f^g    to   an  equation    involving    only 

entire  terms*  Ans,  ad  —  be  +  bdf=z  hdg, 

4.  Reduce  the  equation 

ax      2cH   .    ^         4:hcH      Sa^      2c2 

— +  4a  =  — -—  H 36 

6  tt6  a*^  6^  a 

to  one   involving  only  entire  terms. 

Ans.  a^hx  —  2a^hc^x  ^-  4a*62  _  453^2^  _  5^6  +  2a262c2  -  ^a^h^. 

Secoiid    Transformation, 

79»  The  object  of  the  second  transformation  is  to  change 
any  term  from  one  member  of  an  equation   to   the  other. 

Let  us  take  the  equation 

ax  -\'  b  z=z  d  —  ex, 
[f  we  add  ex    to  both  members,  the  equality  will  not  be  de- 
stroyed (axiom  1),  and  we  shall  have 

ax  +  ex  -\-b  z=d  —  ex  +  cx\ 
or  by  reducing,        ax  +  cx  +  b  =  d. 

Again,  if  we  subtract  b  from  both  members,  the  equality 
will  not  be  destroyed  (axiom  2),  and  we  shall  have,  after 
r^uction, 

ax  +  ex  =  d  —  b. 
Since  we  may  perform  similar  operations  on  any  other  equation, 
we  have,  for  the  change  or  transposition  of  terms,  the  following 

RULE. 

Any  term  of  an  equation  mag  be  transposed  from  one  member 
to  the  other  by  changing  its  sign, 

80»  We  will  now  appiy  the  preceding  principles  to    jhe  solifc 

tion  of  equations  of  the  first  degree. 

For  this  purpose  let  us  assume  the  equation 

a  +  b  ^      ^        a  +  d 

X  ^  d  =  bx . 

c  a 

Clearing  of  fractions,  we  have, 

a{a  +  b)x  —  aed  =  abex  —  c  (a  +  d). 


78  ELEMENTS  OF  ALGEBR^  [CH^P.  IV. 

If,  now,  we  perform  the  operations  indicated  in  both  members, 
we  shall  obtain  the  equation 

<j?x  +  ahx  —  Ojcd  =  abcx  •—  ca  —  cd. 
Transposing   all    the    terms    containing   a;,   to   the   first    member, 
and   all   the   known   terms  to   the   second  member,   we   shall  have, 
a^x  +  abx  —  ahcx  =  acd  —  ac  —  cd. 
Factoring  the  first  member,  we  obtain 

{p^  ■\'  ah  —  abc)  x  =  acd  —  ac  —  cd  : 
If   we    divide    both     members    of    this    equation    by    the    co 
eflacient  of  Xj   we  shall   have 

acd  —  ac  —  cd 
a^  4-  «^  —  cibc 
Any  other  equation  of    the    first    degree  may  be    solved  in   a 
similar  manner : 

Hence,  in  order  to  solve  any  equation  of  the  first  degree, 
we  have  the  following 

RULE. 

I.  Clear  the  equation  of  /inactions,  and  perform  in  both  members 
all  the  algebraic  operations  indicated, 

n.  Transpose  all  the  terms  containing  the  unknown  quantity  to 
the  first  member^  and  all  the  known  terms  to  the  second  member^ 
and  reduce  both  members  to  their  simplest  form. 

III.  Resolve  the  first  member  into  two  factors^  one  of  which  shall 
be  the  unknown  quantity ;  the  other  one  will  be  the  algebraic  sum 
of  its  several  co-efficients. 

IV.  Divide  both  members  by  the  co-efficient  of  the  unknown  quari' 
tity ;  the  second  member  of  the  resulting  equation  will  be  the  re* 
quired  value  of  the  unknown  quantity. 

1.  Take  the  numerical  example 

5X  ^4:X         -,  Q  _    '^  1^^ 

12       T ""       "^  T         W 
Clearing  of  fi-actions 

10a;  —  32a;  —  312  =  21  —  52a;; 


CHAP.  IV.j        EQUATIONS  OF  THE  FIRST  DEGREE.  79 

transposing   and   reducing 

BOX  =  333  : 
Whence,  by  dividing  both  members  of  the  equation  by  30, 

x=n.i. 

If  we  substitute  this  value  of  x,  for  x,  in  the  given  equation^ 
it  will  verify  it,  that  is,  make  the  two  members  equal  to  each 
other. 

Find  the  value  of  a;  in  each  of  the   following 

EXAMPLES. 

1.  3a;  —  2  +  24  =  31.  Ans.  a;  =  3. 

2.  a;  +  18  =  3a;  —  5.  Ans.  x  =  llj. 

3.  6  —  2a;  +  10  =  20  —  3af  —  2.  Ans.  a;  =  2 

4.  a;  +  — a;  +  —  a;  =  11.  Ans,  a;  =  6. 

1  R 

6.   2a; —  a;  +  1  =  5a;  —  2.  Ans.  a;  =  -=-. 

2  T 

^     «       .    «        «       T  A  6    —  3a 

6.   3aa;  +  — 3  =  oa;  —  a.  Ans.  x  = -. 

2  6a  —  26 

.     a;  —  3   .    a;       ^^       a;—  19  . 

*J.   — h  —  =  20 .  Ans.  X  =  23J. 

2  3  2  * 

a;  +  3a;         .       x—  5  .  ^- 

8.  —^  +  —  =  4 —.  Ans.x  =  3^. 

^     ax  —  b   ,    a        hx       hx  —•  a  .  Bb 

9. J-  -—  = — .  Ans.  X  = 


4             3         2  3                                           3a  —  26 

,  ^     3aa;        26a;        ,        .  ^               cdf -{■  4cd 

10. 4  =/.  Ans.  X  =  •-— — rr-. 

c            d                 "^  3ac?  — 26c 

, ,     8aa;  —  6       36  —  c        ,  ,             ^               56  +  96  —  7c 

11.^ 2-  =  *-^       ^•*  = Te^ 

5  3  2         3 


80  ELEMENTS  OF  ALGEBRA.  [CHAP,  IV 

a         0         c         a  oca  —  acd  +  ahd  —  abc 

14.   X — 1 — —  =  a;  +  1.  Ans,  a;  =  6. 

a:        8a;       a;  —  3 

iz>     ^         4a;-2       3a;--l 

16.  2a; —  =  — - — .  Ans,  a;  =  3. 

O  At 

17.  3a;  +  ' — - —  =x  '\-  a,  Ans,  x  r- 


6    +  h' 


ig.  M:i)(^^3,^,Mz:i!_,,  +  «^-5. 


a  —  b  '       a-\-  b  b      ' 

a*  +  ^a%  +  4a262  __  e^S^  +  26* 


-4n3.  a; : 


26  (2a2  +  a6  -  62) 


Problems  giving  rise  to  Equations  of  the  First  Degree^  involv- 
ing but  one  Unhnovju  Quantity, 

81  •  The  solution  of  a  problem,  by  means  of  algebra,  consists 
of  two   distinct   parts — 

1st.  The   statement   of  the  problem ;   and 

2d.   The  solution  of  the  equation. 

We  have  already  explained  the  methods  of  solving  the  equa- 
tion ;  and  it  only  remains  to  point  out  the  best  manner  of  making 
the  statement. 

The  stateme/ot  of  a  problem  is  the  operation  of  expressing, 
algebraically,  the  relations  between  the  known  and  unknown 
quantities  which  enter  it. 

This  part  cannot,  like  the  second,  be  subjected  to  any  well- 
defined  rule.  Sometimes  the  enunciation  of  the  problem  furnishes 
the  equation  immediately ;  and  sometimes  it  is  necessary  to  dis- 
oover,  from  the  enunciation,  new  conditions  from  which  an  equa- 
tion  may  be  formed. 


<3HAP.   IV.         EQUATIONS   OF  THE  FIKST  DEGREE.  81 

The  conditions  enunciated  are  called  explicit  conditions,  and 
those  which  are  deduced  from  them,  implicit  conditions. 

In  almost  all  cases,  however,  we  are  enabled  to  discover  the 
equation  bj  applying  the  following 

RULE. 

Denote  the  unknown  quantity  hy  one  of  the  final  letters  of  tlve 
alphabet,  and  then  indicate^  hy  means  of  algebraic  signs^  the  same 
operations  on  the  Tcnown  and  unknown  quantities^  as  would  he 
necessary  to  verify  the  value  of  the  unknown  quantity^  were  such 
value  known, 

PROBLEMS. 

t 

1.  Find  a  number  such,  that  the  sum  of  one  half,  one  third 
and  one  fourth  of  it,  augmented  by  45,  shall  be  equal  to  448. 

Let  the   required  number   be  denoted  by x. 

Then,  one  half  of  it  will   be  denoted  by — , 

At 

one  third  of  it by---- 


X 
X 


one  fourth  of  it by   -    -    -    - 

and  by  the  conditions,      —  +  —  +  —  +  45  =  448. 
Z         o         4i 

Transposing  -    -    -^  +  -|-  +  4"  =  448  —  45  =  403  ; 
<4         o         4 

bearing  of  fractidtis,    -    -    -    -     Qx  +  Ax  +  ^x  z=z  4836 ; 

reducing, 13a;  =4836; 

iience, a:  =  372. 

Let  lis  see  if  this  value  will  verify  the   equation.     We  have, 

372      372      372 

-2-  +  -3-  +  -^  +  45=  186  +  124  +  93  +  45  =  448. 


82  ELEMENTS   OF  ALGEBRA.  [CHAP.   IV 

2.  What  number  is  that  whose  third  part  exceeds  its  fourtli 
by  16  1 

Let  the  required   aumber  be   denoted  by  x. 

Then,  -^  x    will  denote  the  third  part ; 

o 

and  -T-x    will  denote  the  fourth  part. 

By  the  conditions  of  the  problem, 

-^x--x^    16. 

Cleaiing  of  fractions,  -       4x  —  Sx  =  192; 
reducing, a;  =  192. 

Verification. 
192       192       ,^ 

or,    -    .    -  16  =  16. 

3.  Out  of  a  cask  of  wine  which  had  leaked  away  a  third  part, 
21  gallons  were  afterward  drawn,  and  the  cask  was  then  half 
full :  how  much  did  it  hold  ? 

Suppose  the  cask   to  have  held  x   gallons. 

X 

Then,     -     -    -    -     —    will  denote  what  leaked  away; 

X 

and       -    -    -    .     -_  -f  21    will  denote  what  leaked  out  and 

o 

also  what  was   drawn  out. 

By  the  conditions  of  the  problem, 


i^  ^'=1- 

Clearing  of  fractions,   - 

2x  +  126  =  Sx  ; 

reducing 

-    X    =  -.  126  ; 

dividing  by  —  1    -    - 

X    =       126. 

Verification, 

3+21-    2    , 

or,         .    .    •    - 

63=63. 

CHAP.  IV.l   EQUATIONS  OF  THE  FIRST  DEGREE.  83 

4.  A  fish  was  caught  whose  tail  weighed  9lb. ;  his  head  weighed 
as  much  as  his  tail  and  half  his  body ;  his  body  weighed  as  much 
as  his  head  and  tail  together :   what  was  the  weight  of  the  fish  \ 

Let      -      -        2x    denote  the  weight  of  the  body ; 
then     -      '  9  \-x    will  denote  weight  of  the  head ; 
and  since  the  body  weighed  as  much   as  both  head   and  tail, 
2ir  =  9+    9  +  a; 
or,       -  2ii;  —  ic  =  18  ;  whence,     x  =  18. 

Verification. 

2  X  18  -  18  =  18  ;         or,     18  -^  18. 

Hence,  the  body  weighed -  S6lbs , 

the  head  weighed 21flbs  ; 

the   tail   weighed 9lhs ; 

and  the  whole  fish  -- 12lbs. 

5.  A  person  engaged  a  workman  for  48  days.  For  each  day 
that  he  labored  he  received  24  cents,  and  for  each  day  that  he 
was  idle,  he  paid  12  cents  for  his  board.  At  the  end  of  the  48 
days  the  account  was  settled,  when  the  laborer  received  504 
cents.  Required  the  number  of  tvorking  days^  and  the  nuinber  of 
days  he  was  idle. 

If  these  two  numbers  were  known,  by  multiplying  them  re- 
spectively by  24  and  12,  then  subtracting  the  last  product  from 
the  first,  the  result  would  be  504.  Let  us  indicate  these 
operations   by   means   of  algebraic   signs. 

Let     -      -       X  denote  the  number  of  working  days ; 

^       then      48  —  x  will  denote  the  number  of  idle  days ; 

2A  X  X   z=z    the  amount  earned,  and 
12  (48  —  x)  —    the  amount  paid  for  his  board. 
Then,  from  the  conditions, 

24a; -12  (48-  x)    =    504 
or,  2Ax  -  576  -t-  \2x  =    504. 

Reducing  36rr  ==  504  +  576  =  1080 

whence,  a;  =    30    the  working  days, 

and,  48  —  30  =    18    the  idle  days. 


84'  iJLEJdiENTS  OF  ALGEBRi^  tCHAP.   IV. 

Verification. 

Thirty  days'  labor,  at  24  cents  a  day 
amounts  to 30  X  24  =  720  cts ; 

and  18  days'  board,  at  12  cents  a  day, 
ftraounts  to 18  X  12  =  210  cts  ; 

and  the  amount  received,  is  their  difference,  504  cts. 

The  preceding  is  but  a  particular  case  of  a  general  problem 
which   may   be   enunciated   as   follows. 

Al  person  engaged  a  workman  for  n  days.  For  each  day 
that  he  labored,  be  was  to  receive  a  cents,  and  for  each  day 
that  he  was  idle,  he  was  to  pay  b  cents  for  his  board.  At 
the  end  of  the  time  agreed  upon,  he  received  c  cents.  Re- 
quired  the  number  of  working  days,  and  the  number  of  idle 
days. 

Let     -     -     ic     denote   the  number  of  working   days ;   then, 
71  —  X     will  denote  the  number  of  idle  days ; 

ax     will  denote  the  number  of  cents  he  received;  and 
b  {n  ^  x)   will  denote  the  number  he  paid  out. 
From   the   conditions   of  the   problem, 
ax  —  b  (n  —  x)  =  c. 
Performing  the  indicated  operations,  transposing  and  factoring, 
W0  find, 

{a  +  b)  X  =    c  -{-  on, 

whence,  x  —    "  ^    the  number  of  working  days  ;  and 

an  —  c       ,  T  «  .  Ti      -, 

the  number  of  idle  days. 


~"    a+6' 

If  we  make  ?i  =  48,   a  =  24,   J  =  12   and  c  =:  504,  we  obtain, 
504  +  576 


36 


:  30  ;     and    48  —  a;  =  18  ;    as  before  found. 


0.  A  fox,  pursued  by  a  greyhound,  has  a  start  of  60  leaps. 
He  makes  9  leaps  while  the  greyhound  makes  but  6 ;  but  3 
leaps  of  the  greyhound  are  equivalent  to  7  of  the  fox.  How 
lately  leaps  must  the  greyhound  make  ^o   overtake  the  fox? 


CHAP.   IV.J         EQUATIONS  OF  THE   FIRST  DEGREE,  §5 

Let  us  take  one  of  the  fox  leaps  as  the  unit  of  distance*, 
then,  3   leaps  of  the  greyhound   being   equal   to   7   leaps  of  the 

7 
fox,  one  of  the  greyhound  leaps  will  be  equal  to     — . 

Let  X  denote  the  number  of  leaps  the  greyhound  must  make 
before   overtaking   the   fox. 

Then,  since  the  fox  makes  9  leaps  while   the  hound  makes  G, 
9  3 

r  ^^  2" " 

will  denote  the  number  of  leaps  the  fox  makes  in  the  same  time. 

7 

—  X     will  denote  the  whole  distance  passed  over  by  the  hound ; 
o 

3 

-—-  X     will  denote  the  whole  distance  passed  over  by  the  fox. 

Then,  from   the   conditions  of  the  problem,  , 

I. =^60  +  1.. 

Clearing  of  fractions,  14a;  =  360  +  9a;, 

transposing  and  reducing,      ^x  =  360, 
whence,  x  =    72; 

3  3 

and  —x  =  —    X    72  =  108,    the  nuiuber  of  fox  leaps. 

Verification, 

l2^  =  ,o  +  '-2L^^; 

or,     .    .    -    -  168  =  168. 

7.  A  can  do  a  piece  of  work  alone  in  10  days,  and  B  in  13 
days  :    in  what  time  can   they  do   it   if  they  work   tog^tljor  ? 

Denote  the  number  of  days  by  x,  and  the  work  to  1»  ^me 
ly  1.     Tlien,  in 

1  day  A  can  do  —  of  the  work ;  and  in 

1  day  B  can  do  —  ^^  ^^®  work ;  hen<5e,  In 

X  days  A  can  do  —  of  the   iv^ork ;  and  ii» 

X  days  B  can  do  —  of  the  work : 


/ 


86  ELEMENTS  OF  ALGEBRA.^  [CHAP.   IV. 

Hence,  bj  the   conditions   of  the  question, 

10  ^  13  ~     ' 
clearing  of  fractions,    13a;  +  10:r  =  130  : 
bonce,  a;  =  5^-,     the  number  of  days. 

8:  Divide   $1000   between  A,  B  and  C,  so   that  A  shall   have 
$72   more   than   B,  and   C   $100   more  than  A. 

Ans.  A's  share  =z  $324,   B's   =:  $252,    C's  =z  $424. 


9.  A  and  B  play  together  at  cards.  A  sits  down  with  $S 
and  B  with  $48.  Each  loses  and  wins  in  turn,  when  it  ap- 
pears that  A  has  five  times  as  much  as  B.  How  much  did  A 
win?  Ans.  $26. 

10.  A  person  dying,  leaves  half  of  his  property  to  his  wife, 
one  sixth  to  each  of  two  daughters,  one  twelfth  to  a  servant, 
and  the  remaining  $600  to  tne  poor  :  what  was  the  amount 
of  his   property?  Ans.  |7200. 

11.  A  father  leaves  his  property,  amounting  to  $2520,  to  four 
sons.  A,  B,  C  and  D.  C  is  to  have  $360,  B  as  much  as  G 
and  D  together,  and  A  twice  as  much  as  B  less  $1000  :  how 
much   do  A,  B   and   D   receive? 

Ans.  A   $760,   B   $880,  D   $520. 

12.  An  estate  of  >  $7500  is  to  be  divided  between  a  widow,  two 
sons,  and  three  daughters,  so  that  each  son  shall  receive  twice  as 
much  as  each  daughter,  and  the  widow  herself  $500  more  than 
all  the  children  •  what  was  her  share,  and  what  the  share  of 
each   child  ?  r  Widow's  share,  $4000. 

Ans.   }  Each  son,  $1000. 

(  Each  daughter,  $500. 

13.  A  company  of  180  persons  consists  of  men,  women  and 
children.  The  men  are  8  more  in  number  than  the  women,  and 
iie  children  20  more  than  the  men  and  women  together  :  how 
many  of  each   sort  in   the   company  ? 

Ans.  44  men,  36  women,  100  children. 


CHAP.  IV.J   EQUATIONS  OF  THE  FIRST  DEGREE.  87 

14.  A  father  divides  $2000  among  five  sons,  so  that  each  elder 
should  receive  $40  more  than  his  next  younger  brother :  what  is 
the  share  of  the  youngest?  Ans.   $320. 

15.  A  purse  of  $2850  is  to  be  divided  among  three  persons, 
A,  B  and  C;  A's  share  is  to  be  ^t  ^^  ^'^  share,  and  C  is  to 
have  $300  more  than  A  and  B  together :  what  is  each  one's 
share?  Ans,  A's  $450,  B's  $825,  C's  $1575. 

16.  Two  pedestrians  start  from  the  same  point ;  the  first  steps 
twice  as  far  as  the  second,  but  the  second  makes  5  steps  while 
the  first  makes  but  one.  At  the  end  of  a  certain  time  they  are 
300  feet  apart.  Now,  allowing  each  of  the  longer  paces  to  be  3 
feet,  how  far  will  each   have   traveled  1 

Ans.   1st,  200  feet;  2d,  500. 

17.  Two  carpenters,  24  journeymen,  and  8  apprentices,  re- 
ceived at  the  end  of  a  certain  time  $144.  The  carpenters 
received  $1  per  day,  each  journeyman  half  a  dollar,  and  each 
apprentice   25  cents :  how  many  days  were   they  employed  ? 

Ans.  9  days, 

18.  A  capitalist  receives  a  yearly  income  of  $2940  •  four  fifths 
of  his  money  bears  an  interest  of  4  per  cent.,  and  the  remainder 
of  five   per  cent.  :  how  much   has   he   at  interest  1 

Ans.  $70000. 

19.  A  cistern  containing  60  gallons  of  water  has  three  unequal 
cocks  for  discharging  it ;  the  largest  will  empty  it  in  one  hour, 
the  second  in  two  hours,  and  the  third  in  three  :  in  what  tinye 
will   the   cistern  be   emptied    if  they  all   run   together  ? 

Ans.    32yj  min. 

20.  In  a  certain  orchard  ^  are  apple-trees,  J  peach-trees, 
I  plum-trees,  120  cherry-trees,  and  80  pear-trees :  how  many 
trees   in   the   orchard  ?  Ans.  2400. 

21.  A  farmer  being  asked  how  many  sheep  he  had,  answered 
that  he  had  them  in  five  fields  ^  in  the  1st  he  had  i,  in  the 
2d  "I,  in  the  3d  ^,  in  the  4th  ^,  and  in  the  5th  450  :  how 
many  had  he?  Ans.  1200. 


88  ELEMENTS   OF  ALGEBRA^  [CHAP.   IV, 

22.  My  horse  and  saddle  together  are  worth  $132,  and  the 
horse  is  worth  ten  times  as  much  as  the  saddle  :  v,^hat  is  the 
value  of  the  horse?  A7is,  $120. 

23.  The  rent  of  an  estate  is  this  year  8  per  cent,  greater  than 
it  was  last.     This  year  it  is  $1890 :  what  was  it  last  year  ? 

Ans.  $1750. 

24.  What  number  is  that  from  which,  if  5  be  subtracted,  f  of 
tlie  remainder  will  be  40  ?  Ans.  65. 

25.  A  post  is  -i-  in  the  mud,  J  in  the  water,  and  ten  feet  above 
tiie  water :   what  is   the  whole   length  of  the  post  ? 

Ans,  24  feet 

26.  After  paying  ^  and  \  of  my  money,  I  had  66  guineas  left 
in  my  purse:  how  many  guineas  were   in   it   at  first? 

A91S.    120. 

27„  A  person  was  desirous  of  giving  3  pence  apiece  to  some 
beggars,  but  found  he  had  not  money  enough  in  his  pocket  by  8 
pence ;  he  therefore  gave  them  each  two  pence  and  had  3  pence 
remaining:  required  the  number  of  beggars.  Ans.  11. 

28.  A  person  in  play  lost  ^  of  his  money,  and  then  won  3 
jjliillings ;  after  which  he  lost  ^  of  what  he  then  had ;  and  this 
done,  found  that  he  had  but  12  shillings  remaining :  what  had 
he  at  first?  Ans.  20s. 

29.  Two  persons,  A  and  B,  lay  out  equal  sums  of  money  in 
trade ;  A  gains  $126,  and  B  loses  $87,  and  A's  money  is  now 
double  B's  :  what  did  each  lay  out  ?  Ans.  $300. 

30.  A  person  goes  to  a  tavern  with  a  certain  sum  of  money 
ill  his  pocket,  where  he  spends  2  shillings ;  he  then  borrows 
as  much  money  as  he  had  lefl,  and  going  to  another  tavern, 
he  there  spends  2  shillings  also  ;  then  borrowing  again  as 
much  money  as  was  left,  he  went  to  a  third  tavern,  where^ 
likewise,  he  spent  2  shillings  and  borrowed  as  much  as  he 
had  left ;  and  again  spending  2  shillings  at  a  fourth  tavern, 
he   then   had   aothing  remaining;.     What  had  he  at  first  ? 

Ans.  Ss.  9d, 


CHAP.   III.]        EQUATIONS  OF  THE   FIEST  DEGREE.  89 

31.  A  farmer  bought  a  basket  of  eggs,  and  offered  them  at  7 
cents  a  dozen.  But  before  he  sold  any,  5  dozen  were  broken 
by  a  careless  boy,  for  which  he  was  paid.  He  then  sold  the  re- 
mainder at  8  cents  a  dozen,  and  received  as  much  as  he  would 
have  got  for  the  whole  at  the  first  price.  How  many  eggs  had 
he  in  his  basket?  Ans,  40  dozen. 

Equations   of  the  First  Degree  involving  more   than   one 
Unknown    Quantity. 

82*  If  we  hare   an  equation  between  two  unknown  quantities, 

we   may  find   an   expression   for   one   of   them   in   terms   of   the 

other    and   known   quantities ;   but    the    value   of   this    unknown 

quantity    could    only   be   determined    by    assuming   a   value   for 

the   second.     Thus,  from   the   equation, 

a;  +  2?/  =  4, 
we   may   deduce 

x  =  4.    -2y, 

but  cannot  find   a   value   for   x   without   assuming   one   for   y. 

If,  however,  we  have  another  equation  between  the  two  un 
known  quantities,  the  values  of  these  quantities  being  the  same 
in  both,  we  may  find,  as  before,  an  expression  for  x  in  termxS 
of  y,  and  this  expression  placed  equal  to  the  one  already 
found,  will  give  an  equation  containing  but  one  unknown  quan- 
tity.    Let  us   take 

a;  +  3?/  =  5, 
from  which  we  find 

X  =:    5  —  3?/. 

If  we   place  this    expression   equal   to   that  before  found,  we 

deduce   the   equation 

4  -  2y  =  5  -  3y, 
fi'om   the    solution   of  which  we  find,    y  =  1. 

This  value  of  y,  substituted  in  either  of  the  given  equations, 
gives   a;  =  2  :  hence, 

a;  =  2   and   y  =  1    satisfy  both  equations. 

We  see  that  in  order  to  find  determinate  values  for  two 
unknown  quantities,  we  must  have  two  independent  equations* 
Simultaneous  equations  are  those  in  which  the  values  of  the 
»'iiknown  quantities  are  the  same  in   them  all  at  tie  same  time 


90  ELEMENTS    OF  ALGEBRA.  [CHAP.   IV. 

111  the  same  manner  it  may  be  shown  that  to  determine  the 
values  of  three  unknown  quantities,  we  must  have  three  equa- 
tions ;  and  generally,  to  determine  the  values  of  n  unknown 
quantities  we  must  have   qi    equations. 

Elimination. 

83t  JElimination  is  the  operation  of  combining  several  equations 
involving  several  unknown  quantities^  and  deducing  therefrom  a  less 
number  of  equations  involving  a  less  number  of  unknown  quantities. 

There   are   three  principal   methods   of  elimination : 

1st.  By   addition   or   subtraction. 

2d.    By   substitution. 

3d.    By   comparison. 

We   shall   explain  these  methods   separately. 

JElimination  by  Addition  or  Subtraction. 

84»  Let  us   take  the   two   equations 
4:X  —  by  =    5, 
So:  +  2y  =  21. 
If  we   multiply   both    members   of   the   first    equation    by  2, 
the   co-efficient   of  y   in   the   second,  and   both   members   of  the 
second  equation  by  5,  the  co-efficient  of  y  in  the  first,  we  obtain, 
Sx  —  lOy  =    10, 
15^  +  lOy  =  105  ; 
in  which  the  co-efficients  of  y  are  numerically  the  same  in  both. 
If,  now,  we  add  these  equations  member  to  member,  we  find 

2Sx  =  115. 
In   this   case   y   has   been    eliminated  by   additior^. 
Again,  let   us   take   the   equations 

2x  +  Sy=:  12, 
Sx  +  4:y=  17. 
If    we    multiply   both    members   of   the   first    equation   by   8, 
the   co-efficient   of    x    in   the    second,    and   multiply   both   mem- 
bers  of  the   second   equation   by  2,  the   co-efficient  of  x   in   the 

first;  we   shall   have, 

6x  +  9y  =  36, 

6a;  +  8y  =  34  ; 


CHAP.    IV.]        EQUATIONS  OF  THE  FIKST   DEGKEE.  91 

in  which  the  co-efficients  of  x  are  the  same  in  both.  If,  now, 
we  subtract  the  second  equation  from  the  first,  member  from 
member,  we  find, 

y  =  2. 

Here,    x   has  been  eliminated  Z>y  subtraction. 

In  a  similar  manner  we  may  eliminate  one  unknown  quantity 
Detween  any  two  equations  of  the  first  degree  containing  any 
number  of  unknown  quantities.  The  rule  for  elimi^Ation  by 
addition  and  subtraction  may  be  simplified  by  using  the  least 
common  multiple.  Hence,  for  elimination  by  addition  or  sub- 
traction,  we  have   the  following 

RULE. 

Prepare  the  two  equations  in  such  a  manner  that  the  co-efficients 
of  the  quantity  we  wish  to  eliminate  shall  be  numerically  equal 
in  both :  then^  if  the  two  co-efficients  have  contrary  signs ^  add  the 
equations^  member  to  member ;  if  they  have  the  same  sign,  sub- 
tract them  member  from  member,  and  the  resulting  equation  will 
be  independent  of  that  quantity. 

Elimination  hy  Substitution. 

85 •  Let  us  take  the   equations, 

5a;  +  7y  =  43,     and     llx -[- ^y  —  Q9f, 

Find,  from   the  first   equation,  the  value  of  x  in    terms  of  y, 

w^hich  is, 

43-7y 

="  =  — ^- 

Substitute   this  value  for    x    in   the   second   equation,  and  we 

shall   have 

11  X  (43-72/)      ^       ' 

"^-^ ^  +  9y  =  69;     or,. 

reducing,     -     -     -     473  —  lly  +  45?/  =  345. 

In  a  similar  manner  we  may  eliminate  one  unknown  quantity 
between  two  equations  of  the  first  degree  containing  any  numbei 
of  unknown  quantities. 

Hence,  for  eliminating  by  substitution,  wo  have  the  followinsf 


92  ELEMENTS   OF  ALGEBRA.  [CHAP.   IV. 

RULK 

Find  from  one  equation  the  value  of  the  unknown  quantity  to 
he  eliminated  in  terms  of  the  others :  substitute  this  value  in  the 
other  equation  for  the  unknown  quantity  to  he  eliminated^  and  the 
resulting   equation  will  he  independent  of  that  quantity. 

Elimination   hy    Comparison, 

86*  Let  us  take  the   equations, 

5a:  4-  7y  =  43, 

lla;  +  9y  ==69. 

Finding  the  value  of  x  in  terms  of  y,   from    both   equations 

we  have, 

43 -7y 

_  69  —  9y 
a?-       ^^      . 

If,  now,  we  place  these  values  equal  to  each  other,  we  shall  have, 
43  -  7y  _  69  -  9y 
5       ~       11       ' 
reducing,     -.    -     -     473  —  77y  =:  345  —  45?/. 

Here,  x  has  been  eliminated.  Generally,  if  we  have  two 
equations  of  the  first  degree  containing  any  number  of  unknown 
quantities,  any  one  of  them  may  be  eliminated  by  the  following 

RULE. 

Find  the  value  of  the  quantity  we  wish  to  eliminate^  in  terms 
of  the  others^  from  each  equation^  and  then  place  these  values 
equal  to  each  other :  the  resulting  equation  ivill  he  independent 
of  the  quantity   whose   values   were  found. 

The  new  equations  which  arise,  from  the  two  last  method? 
of  elimination,  contain  fractional  terms.  This  inconvenience  is 
avoided  in  the  first  method.  The  method  hy  substitution  is, 
however,  advantageously  employed  whenever  the  co-efficient  of 
either  of  the  unknown  quantities  in  one  of  the  equations  is  equal 
to   1,   because   then   the   inconvenience   of  which   we    have    jusl 


CHAP.   IV.J         EQUATIOxSS  OF    THE   FIRST  DEGREE.  93 

spoken  doe:  not  oocur.  We  shall  sometimes  have  occasion  to 
employ  this  method,  but  generally  the  method  by  addition  and 
subtraction  is  preferable.  When  the  co-efficients  are  not  too 
great,  the  addition  or  subtraction  may  be  performed  at  the 
same  time  with  the  multiplication  that  is  made  to  render  the 
oo-efficients  of  the  same  unknown  quantity  equal  to  each  other. 

There  is  also  a  method  of  elimination  by  means  of  the 
greatest  common  divisor,  which  will  be  explained  in  its  appro- 
priate place. 

87«  Let  us  now  consider  the  case  of  three  equations  involving 
three  unknown   quantities. 

rbx  —  Q>y  +  4:Z=z  15; 
Take  the  equations,      •<  7a;  +  4y  —  3^  =  19, 

(2x-\-    y  +  6.'2i=46.  _ 

To  eliminate  z  from  the  first  two  equations,  multiply  the  first 
equation  by  3  and  the  second  by  4 ;  and  since  the  co-efficients 
of  z  have  contrary  signs,  add  the  two  results  together :  this  gives 
a  new  equation,        ...        -  43aj  —  2y  =  121.'' 

Multiplying  both  members  of  the   second 
-equation  by  2,  a  factor  of  the  co-efficient  of 
z  in   the   third   equation,  and   adding  them, 
member  to  member,  we  have  -        -        -         16a:  +  9y=    84.^ 

The  question .  is  then  reduced  to  finding  the  values  of  x  and  y, 
which  will  satisfy  these   new  equations. 

Now,  if  the  first  be  multiplied  by  9,  the  second  by  2,  and 
the   results   be   added   together,  we  find 

419a;  =  1257,     whence     a;  =  3, 
By  means  of  the   two   equations  involving  x  and  y,  we  may 
determine  y  as  we  have  determined  x ;  but  the  value  of  y  may 
DC   determined    more    simply,    since    by   substituting   for    x    its 
value  found  above,  the  last  of  the  two   equations  becomes, 
48  +  9y  =  84,     whence    y  =  4. 
In  the   same   manner,  by  substitnting  the  values  of  x  and  y, 
Uie  first  of  the   three   proposed   equations   becomes, 
15  —  24  +  4:Z  =  15,     whence    ^  =  6. 


;  94  ELEMENTS  OF  AL(tii.BKA.  [CHAP.   lY. 

If  we  have  a  group  of  m  simultaneous  equations  ccntaining  m 
unknown  quantities,  it  is  evident,  from  principles  already  ex- 
plained, that  the  values  of  these  unknown  quantities  may  be 
found  by  the  following 

RULE. 

L  Combine  one  of  the  m  equations  with  each  of  the  m  —  1  others, 
separately,  eliminating  the  sam^e  unknown  quantity  ;  ihere  will  result 
m  —  1    equations   containing   m  —  1    unknown   quantities, 

II.  Combine  one  of  these  with  each  of  the  m  —  2  others,  sepa- 
rately, eliminating  a  second  unknown  quantity ;  there  will  result 
m  —  2    equations   containing   m  —  2    unknown   quantities. 

III.  Continue  this  operation  of  combination  and  elimination  till 
we  obtain,  finally,  one  equation  containing  one  unknown  quantity. 

IV.  Find  the  value  of  this  unknown  quantity  by  the  rule  for 
solving  equations  of  the  first  degree  containing  one  unknown  quan- 
tity :  substitute  this  value  in  either  of  the  two  preceding  equations 
containing  two  unknown  quantities,  and  determine  the  value  of  a 
second  unknown  quantity  :  substitute  these  two  values  in  either  of 
the  three  equations  involving  three  unknown  quantities,  and  so  on 
till   we  find   the   values   of  them   all. 

It  often  happens  that  some  of  the  proposed  equations  do  nc 
contain  all  the  unknown  quantities.  In  this  case,  with  a  littk 
address,  the  elimination  is  very  quickly  performed. 

Take  the  four  equations  involving  four  unknown  quantities, 

2.r  -  3y  4-  2^  =  13         -     (1)  4y  -f  2^  =  14    -     (3). 

4.u-2x  =  ^0         -     (2)  5z/-f3zt  =  32     -     (4). 

By  examining  these  equations,  we  see  that  the  elimination  of 
z  in  equations  (1)  and  (3),  will  give  an  equation  involving  r 
and  y  ;  and  if  we  eliminate  u  in  the  equations  (2)  and  (4),  we 
shall  obtain  a  second  equation,  involving  x  and  y.  In  the  first 
place,  the  elimination  of  ^,  in  (1)  and  (3)  gives  Ty  —  2a;  =  1  -  (5), 
that  of  u,  in  (2)  and  (4),  gives      -        -      20y  +  6a;  =  3^  -  (6). 

From  (5)  'and  (6)  we  readily  deduce  the  values  oi  y  ~  1  and 
r  =  3 ;  and  by  substitution  in  (2)  and  (3),  we  also  find  u  ^^ 
and  2?  =  5. 


CHAP.   ly.J        EQUATIONS  OF  THE   FIRST  DEGREE. 

EXAMPLES. 


95 


1.  Given    2^5  +  3y  =  16,  and    Sx  ^2y=zll    to  find  the  values 
of  X  and  y.  A71S,  a;  =  5,    y  =  2. 

^    ^.  2x      Sy       9  ^    3a;   ,2y       61       ^      ^    -      - 

2.  Given    y  +  f  =  25,  and     4"  +  f  =  t^o     to    find     the 


values  of  x  and  y. 


2'    ^        3 


3.  Given    -^-{-ly  z=z  99,    and    -^  +  7a;  =  51  to  find  the  values 


of  a;  and  y. 


Ans.  a;  =  7,    ?/  =  14. 


4.  Given    |-12=:  1- +  8,  and  ^^  +  |.-8=fcf +27 
2  4  5  t>  4 

to  find  the  values  of  x  and  y,  Ans,  a;  =  60,    y  =  40. 

a;+     2/+      ^  =  29 


5.  Given 


x+   2y+    3^  =  62 


U'+i 


to  find  a;,  y,  and  2. 


3y  +  -4-  =  10 


6.  Given 


Ans,  X  =  S,    y  z=z  9,  z  =z  12. 

2x+    4y—    Sz=z  22 

to  tind  X,  y,  and  e, 
72/- 

^/i5.  a;  =  3,    y  =  7,    2  =  4. 


/  2x+  4y—  Sz=z22  ^ 
3  4a;  -  2y  +  50  =  18  V 
I     6a;  +    7y  -      2  =  63  ) 


7   Given 


8.  Given 


«+-^y  +— ^  =  32 

'o'^+'j-y +  -^z  =  l^  )-     to  find  X,  y,  and  «. 

^|^  +  yy  +  -^.  =  12 

^715.  a;  =  12,    y  =  20,    2;  =  30. 

f  7a;  -  2^  +  3w  =  17  ^ 
4y-  22+  ^=11 
5y  —  3a;  —  2w  =  8 
4y  —  3t^  +  2^  =  9 
32+  8t^  =  33 
.^w*.  a;  =  2,    y  =  4,    2  =  3,    w  =  3,    <  =  1 


>.      to   find   a;,   y,  0,   t«, 
and  t. 


96  ELEMENTS  OF  ALGEBRA.  [CHAP.  IV. 

PROBLEMS     GlVma    RISE    TO    SIMULTANEOUS    EQUATIONS    OF   THE   FIRST 

DEGREE, 

1.  What  fraction  is  that,  to  the  numerator  of   which,  if  1   be 
ftdded,  its  value  will   be   one   third,   but    if    1    be   added  to  its 
ienominator,  its  value  will   be   one  fourth? 
Let    X   denote   the  numerator,  and 

y  the  denominator. 
From   the   conditions   of  the  problem, 
x-\-l        1 


y 

-  3' 

X 

1 

y  +  1" 

Gearing  of  fractions, 

the  first 

equation 

gives, 

3^  +  3  = 

=  y, 

and  the  2d, 

^x 

=  2/  +  l. 

Whence,  by  eliminating   y, 

0^-3  = 

=  1, 

and 

X  - 

=  4. 

Substituting,  we  find, 

y  = 

=  15; 

4 

and  the  required  fraction  is    — . 

2.  To  find  two  numbers   such   that  their  sum   shall  be  equal 
to   a  and  their  difference  equal  to   b. 
Let    X    denote  the  greater  number,  and 

y    the  lesser  number. 
From  the   conditions  of  the  problem, 
x  +  y  =a, 
X  —  y  =  b. 
Eliminating  y  by  addition, 

2x  =  a  +  b, 
a         h 


By  substitution, 


2    '    2' 


_    a         h 

^"^  y  2"; 


CHAl'.   IV.J        EQUATIONS   OF  THE   FIRST  DEGREE,  97 

3.  A  person  possessed  a  capital  of  30000  dollars,  for  which 
he  drew  a  certain  interest  per  annum ;  but  he  owed  the  sum 
of  20000  dollars,  for  which  he  paid  a  certain  interest.  ^  The 
interest  that  he  received  exceeded  that  which  he  paid  by  800 
dollars.  Another  person  possessed  35000  dollars,  for  which 
he  received  interest  at  the  second  of  the  above  rates ;  but  he 
owed  24000  dollars,  for  which  he  paid  interest  at  the  first 
of  the  above  rates.  The  interest  that  he  received  exceeded 
that  which  he  paid  by  310  dollars.  Required  the  two  rates 
of  interest. 

Let     X    denote  the  first  rate,  and 

y    the  second  rate. 

Then,  the  interest  on  $30000  at  x  per  cent,  for  one  year  will  be 

$30000a;  ^^^^ 

^^Q  or     $300ar. 

The  interest  on  $20000  at   y   per  cent,  for  one  year  will  be 

$20000y  ^^^^ 

-^     or     $200y. 

Hface,  from  the  first  condition  of  the  problem, 
300a;  —  «00y  =  800 ; 
or,       -    -    -    -         3a;—      2y=      8     -     -     -     (1). 
In  like  manner  from  the  second  condition  of  the  problem  we  find 

35y-    24ar=r    31     .     -     -     (2). 
Combining   equations  (1)  and  (2)  we  find, 
y  =  5     and    a;  =  6. 

Hence,  the  first  rate  is  6  per  cent,  and  the  second  rate  5 
per  cent. 

Verification. 

$30000,  placed  at  6  per  cent,,  gives  $300  X  6  =  $1800. 

$20000         do         5  do  $200  X  5  =  $1000.     * 

And  we  have  1800  —  1000  =  800. 

The  second  condition  can  be  verified  in  the  same  manner. 

4.  There  are  three  ingots  formed  by  mixing  together  three 
metals  in  difierent  proportions. 

7 


98  ELEMENTS   OF  ALGEBRA.^  [CHAP.   IV 

One  pound  of  the  first  contains  7  ounces  of  silver,  3  ounces 
of  copper,  and  6  ounces  of  pewter. 

One  pound  of  the  second  contains  12  ounces  of  silver,  3  oun(5es 
of  copper,  and  1  ounce  of  pewter. 

Due  pound  of  the  third  contains  4  ounces  of  silver,  7  ounces 
of  copper,  and  5  ounces  of  pewter. 

It  is  required  to  form  from  these  three,  1  pound  of  a  fourth 
ingot  which  shall  contain  8  ounces  of  silver,  3f  ounces  of  cop- 
per, and  4|  ounces  of  pewter. 

l^et     X    denote  the  number  of  ounces  taken  from  the  first. 
y    denote  the  number  of  ounces  taken  from  the  second 
z    denote  the  number  of  ounces  taken  from  the  third. 

Now,  since  1  pound  or  16  ounces  of  the  first  ingot  contains  7 

ounces  of  silver,  one   ounce  will   contain   t-t    of  7  ounces :  that 

16 

is,    — -    ounces ;   and 
16 

Ix 
X  ounces  will  contain    —7    ounces  of  silver, 
Id 

y   ounces  will  contain  —~  ounces  of  silver, 

z    ounces  will  contain   —7    ounces  of  silver. 
lb 

But  since  1  pound  of  the  new  ingot  is  to  contain  8  ounces  of 

tiilver,  we   have 

Ix       I2y      4g  __ 

16  "^  16  "^  16         ' 

or,  clearing  of  fractions,  we  have, 
for  the  silver,  7x  +  12y  +  4z  =  128  ; 

for  the  copper,  Sx -{-    Sy  +  7z=    60  ; 

and  for  the  pewter,       6x -\-      y  +  5z  =:    68. 

Whence,  finding  the  values  of  x,   y  and  z,  we  have 
*         X  =zS,  the  number  of  ounces  taken  from  the  first. 
'  2/  =  5     "  "  "  «         «        "    second. 

2  =  3     «  "  "  "         "        "    third. 

5.  What  two  numbers  are  they,  whose  sum  is  33  and  whose 
difference  is  7?  Ans,  20  and   13. 


CHAP.   ly.J       EQUATIONS  OF  THE   FIRST  DEGREE.  99 

6.  Divide  the  number  75  into  two  such  parts,  that  three  times 
the  greater  may  exceed  seven  times  the  less  by  15. 

Ans.  54  and  21. 

7.  In  a  mixture  of  wine  and  cider,  ^  of  the  whole  plus  25 
gallons  was  wine,  and  ^  part  minus  5  gallons,  was  cider  ;  how 
many  gallons  were  there  of  each  ? 

Ans.  85  of  wine,  and  35  of  cider. 

8.  A  bill  of  £120  was  paid  in  guineas  and  moidores,  and  the 
number  of  pieces  of  both  sorts  that  were  used  was  just  100 ;  if 
the  guinea  were  estimated  at  21s.,  and  the  moidore  at  275.,  how 
many  were   there   of  each*?  Ans.  50. 

9.  Two  travelers  set  out  at  the  same  time  from  London  and 
York,  whose  distance  apart  is  150  miles ;  they  travel  toward 
each  other;*  one  of  them  goes  8  miles  a  day,  and  the  other 
7;   in   what  time  will   they  meet?  Ans.  In  10  days. 

10.  At  a  certain  election,  375  persons  voted  for  two  candi 
dates,  and  the  candidate  -  chosen  had  a  majority  of  91  ;  how 
many  voted  for   each? 

Ans.  233  for  one,  and  142  for   the   other. 

11.  A's  age  is  double  B's,  and  B's  is  triple  C's,  and  the  sum 
of  all  their  ages  is  140 ;  what  is  the  age  of  each  1 

Ans.  A's  =  84,  B's  =  42,  and   C's  =  14. 

12.  A  person  bought  a  chaise,  horse,  and  harness,  for  £60 ; 
the  horse  came  to  twice  the  price  of  the  harness,  and  the  chaise 
to  twice  the  price  of  the  horse  and  harness ;  what  did  he  give 
for  each  1  /  £13     6s.  Sd.     for  the  horse. 

Ans.    <  £  6  13s.  4d.     for  the  harness. 
(  £40  for  the  chaise. 

13.  A  person  has  two   horses,  and  a  saddle  worth  £50 ;  now, 
if  the  saddle  be  put  oh  the  back  of  the  first  horse,  it  will  make 
his  value  double  that  of  the  second ;  but  if  it  be  put  en  the  back 
of  the   second,  it  will   make  his  value   triple   that  of  the   first 
what  is  the  value  of  each   horse  ? 

Ans.  One  £30,  and  the  other  £40. 


100  ELEMENTS   OF  ALGEBRA.^  [CHAP.    IV 

14.  Two  persd  as,  A  and  B,  have  each  the  same  income.  A 
saves  J-  of  his  yearly ;  but  B,  by  spending  £50  per  annum  more 
than  A,  at  the  end  of  4  years  finds  himself  £100  in  debt ;  what 
is  the   income  of  each*?  Ans',  £125. 

15.  To  divide  the  number  36  into  three  such  parts,  that  J  of 
the  first,  J  of  the  second,  and  ^  of  the  third,  may  be  all  equal 
to    each   other.'  Ans.  8,  12,  and  16. 

1C>  A  footman  agreed  to  serve  his  master  for  £8  a  year  and 
d  livery,  but  was  turned  away  at  the  end  of  7  months,  and  re 
reived  only  £2  135.  4c?.  and  his  livery ;  what  was  its  value  ? 

Ans.  £4  16s. 

17.  To  divide  the  number  90  into  four  such  parts,  that  if  the 
first  be  increased  by  2,  the  second  diminished  by  2,  the  third 
multiplied  by  2,  and  the  fourth  divided  by  2,  the  sum,  difference, 
product,  and  quotient,  so  obtained,  will  be  all  equal  to  each  other. 

Ans.  The  parts  are  18,  22,  10,  and  40. 

18.  The  hour  and  minute  hands  of  a  clock  are  exactly  together 
at  12  o'clock ;  when  are  they  next  together  ? 

Ans.    1  h.    5^j  mm. 

19.  A  man  and  his  wife  usually  drank  out  a  cask  of  beer  in 
12  days ;  but  when  the  man  was  from  home,  it  lasted  the  woman 
30  days;  how  many  days  would  the  man  be  in  drinking  it 
alone  ?  Ans.  20  days. 

20.  If  A  and  B  together  can  perform  a  piece  of  work  in  8 
days,  A  and  C  together  in  9  days,  and  B  and  C  in  10  days; 
how  many  days  would  it  take  each  person  to  perform  the  same 
work  alone  ?  Ans.  A  14ff  days,  B  17|-f ,  and  C  23/5-. 

21.  A  laborer  can  do  a  certain  work  expressed  by  a,  in  a  time 
expressed  by  ^;  a  second  laborer,  the  work  c  in  a  time  (Z;  a 
third,  the  work  e  in  a  time  /.  Required  the  time  it  would  take 
the   three   la"borers.  workir^   together,  to   perform   the  work  g. 

'        '  Ans. ^— 


adf+bcf+bde 


CHAP.   IV.J        EQUATIONS  OF  THE   FIEST  EEGR|:E.  101 

22.  If  32  pounds  of  sea  water  contair  1  poind  of  salt,  how 
much  fresh  water  must  be  added  to  these  32  pounds,  in  order 
that  the  quantity  of  salt  contained  in  32  pounds  of  the  new  mix- 
ture   shall  be  reduced   to   2   ounces,  or  ^  of  a  pound? 

Ans.  224  lbs. 

23.  A  number  is  expressed  by  tnree  figures ;  the  sum  of  these 
figures  is  11 ;  the  figure  in  the  place  of  units  is  double  that  in 
the  place  of  hundreds;  and  when  297  is  added  to  this  number, 
the  sum  obtained  is  expressed  by  the  figures  of  this  number  re- 
versed.    What  is   the   number  ?  Ans.   326. 

24.  A  person  who  possessed  100000  dollars,  placed  the  greater 
part  of  it  out  at  5  per  cent,  interest,  and  the  other  part  at  4  per 
cent.  The  interest  which  he  received  for  the  whole  amounted 
to  4640   dollars.     Required   the   two   parts. 

„  Ans.  $64000   and   $36000. 

25.  A  person  possessed  a  certain  capital,  which  he  placed  out 
at  a  certain  interest.  Another  person  possessed  10000  dollars 
more  than  the  first,  and  putting  out  his  capital  1  per  cent,  more 
advantageously,  had  an  income  greater  by  800  dollars.  A  third, 
possessed  15000  dollars  more  than  the  first,  and  putting  out  his 
capital  2  per  cent,  more  advantageously,  had  an  income  greater 
by  1500  dollars.  Required  the  capitals,  and  the  three  rates  of 
mterest. 

Sums  at  interest,       $30000,      $40000,      $45000. 

Rates  of  interest,  4  5  6         per  cent. 

26.  A  cistern  may   be  filled  by   three  pipes.   A,   B,   C.     By 

the   two   first  it   can   be   filled   in   70   minutes;  by  the  firsc  and 

third   it   can  be  filled  in   84   minutes ;   and   by    the   second  and 

third   in   140   minutes.     What  time  will    each  pipe   take   to   do 

it  in  ?     What  time  will  be  required,   if   the    three    pipes  run 

together  ^ 

/A  in  105  minutes. 

Ans,  <B  in  210  minutes. 

(  C  in  420  minutes. 
All   will   fill  it  in   one  hour. 


102  ELEMENTS   OF  ALGEBRA.  •  [CHAP.   IV 

27.  A,  has  3  purses,  each  containing  a  certain  sum  of  money 
If  $20  be  taken  out  of  the  first  and  put  into  the  second,  it 
will  contain  four  times  as  much  as  remains  in  the  first.  If  $60 
be  taken  from  the  second  and  put  into  the  third,  then  this  will 
contain  If  times  as  much  as  there  remains  in  the  second.  Again, 
if  .$40  be  taken  from  the  third  and  put  into  the  first,  then 
the  third  will  contain  2|  times  as  much  as  the  first.  What 
were  the  contents  of  each  purse  1  /1st.  $120. 

Ans.    V^d.    $380. 
$500. 


/  1st. 
]2d. 
I  3d. 


28.  A  banker  has  two  kinds  of  money ;  it  takes  a  pieces  of 
the  first  to  make  a  crown,  and  b  of  the  second  to  make  the 
same  sum.  Some  one  ofiers  him  a  crown  for  €  pieces.  How 
many  of  each   kind  must   the  banker  give   him? 

Ans.  1st  kind,   "±=1^.    2d  kind,    ^J^. 
a  —  0  a  —  0 

29.  Find  what  each  of  three  persons.  A,  B,  C,  is  worth, 
knowing,  1st,  that  what  A  is  worth  added  to  I  times  what  B 
and  C  are  worth,  is  equal  to  p  ;  2d,  that  what  B  is  worth 
added  to  m  times  what  A  and  C  are  worth,  is  equal  to  q ; 
3d,  that  what  C  is  worth  added  to  n  times  what  A  and  B  are 
worth,  is  equal  to  r. 

If  we  denote  by  s  what  A,  B,  and  C,  are  worth,  we  intro- 
duce an  auxiliary  quantity,  and  resolve  the  question  in  a  veiy 
simple  manner. 

30.  Find  the  values  of  the  estates  of  six  persons.  A,  B,  C,  D, 
E,  F,  fi.^om  the  following  conditions :  1st.  The  sum  of  the  estates 
of  A  and  B  is  equal  to  a ;  that  of  C  and  D  is  equal  to  b ;  and 
that  of  E  and  F  is  equal  to  c.  2d.  The  estate  of  A  is  worth  m 
times  that  of  C ;  the  estate  of  D  is  worth  n  times  that  of  E,  and 
the  estate  of  F  is  worth  p  times  that  .of  B. 

This  problem  may  be  solved  by  means  of  a  single  equation^ 
involving  but  one  unknown  quantity. 


CHAP.   IV.l        EQUATIONS  CF  THE   FIRST  DEGREE.  108 

Of  Indeterminate  Equations  and  Indeterminate  Problems, 

88 •  An  equation  is  said  to  be  indeterminate  "^rhen  it  may  be 
satisfied  for  an  infinite  number  of  sets  of  values  of  the  unknown 
quantities  which  enter  it. 

Every  single  equation  containing  two  unknown  quantities  is  indo' 
terminate, 

YoT  example,  let  us  take  the  equation 
5a;  — 3y  =  12, 

12  +  Sg 


WllCU 

vc,         - 

■    *-        5 

• 

ow 

,  by 

making 

successively, 

y 

=  1, 

2, 

3, 

4, 

5, 

X 

=  3, 

18 
5' 

21 
5' 

24 
5' 

* 

27 
5' 

6,     &c^ 

6,     &c., 

and  any  two   corresponding  values  of  x,  y,  being   substituted   iii 
the  given  equation, 

6x  —  Sg=z  12, 
will  satisfy  it:  hence,  there  are  an  infinite  number  of  values  for 
X  and  y  which  will   satisfy  the   equation,  and   consequently  it  is 
indeterminate;  that  is,  it  admits  of  an  infinite  number  of  solutions. 

If  an  equation  contains  more  than  two  unknown  quantities,  we 
may  find  an  expression  for  one  of  them  in  terms  of  the  others. 

If,  then,  we  assume  values  at  pleasure  for  these  others,  we 
can  find  from  this  equation  the  corresponding  values  of  the  first ; 
and  the  assumed  and  deduced  values,  taken  together,  will  satisfy 
the  given  equation.     Hence, 

Every  equation  involving  more  taan  one  unknown  quantity  is 
indeterminate. 

In  general,  if  we  have  n  equations  involving  more  than  n 
unknov/n  quantities,  these  equations  are  indeterminate ;  for  we 
may,  by  combination  and  elimination,  reduce  them  to  a  single 
equation  containing  more  than  one  unknown  quantity,  which  we 
have  already  seen  is  indeterminate. 

If,  on  the  contrary,  we  have  a  greater  number  of  equatious 
than   we  have   unknown  quantities,  they   cannot  all  be  satisfied 


104  ELEMENTS   OF    ILGEBRA^  [CHAP.   IV 

unless  some  of  them  are  dependent  upon  the  others.  If  we 
combine  them,  we  may  eliminate  all  the  unknown  quantities,  and 
the  resulting  equations,  which  will  then  contain  only  known 
i^uantities,  will  be  so  many  equations  of  condition^  which  must  be 
satisfied  in  order  that  the  given  equations  nay  admit  of  solution. 
For  example,  if  we  have 

X  -^  y  =a, 

xy        —d', 

we  may  combine  the  first  two,  and  find, 

a    ^     c  _  a        c 

x=-+-     and      ^=2-2-; 

and  by  substituting  these  in  the  third,  we  shall  find 

which  expresses  the  relation  between  a,  c  and  c?,  that  must  exist, 
in  order  that  the  three  equations  may  be  simultaneous. 

88*.  A  Prohlem  is  indeterminate  when  it  admits  of  an  infinite 
number  of  solutions.  This  will  always  be  the  case  wh^n  its 
enunciation  involves  more  unknown  quantities  than  there  are 
given  conditions  ;  since,  in  that  case,  the  statement  of  the  problem 
will  give  rise  to  a  less  number  of  equations  than  there  are 
unknown  quantities. 

1st.  Let  it  be  required  to  find  two  numbers  such  that  5 
times  the  first  diminished  by  3  times  the  second  s^>all  be 
equal  to   12. 

If  we  denote  the  numbers  by  x  and  y,  the  condition'j  cf  the 
problem   will   give   the   equation 

dx  —  Sy=z  12, 
w^hich    we    have    seen    is    indeterminate : — Hence,  the    '^.o'tlem 
admits  of  an  infinite  number  of  solutions,  or  is  indete^r''n».te. 

2.  Find  a  quantity  such  that  if  it  be  multiplied  by  a  and 
the  product  increased  by  b,  the  result  will  be  equal  to  c  time* 
the  quantity   increased  by  d 


CHAP.   IV.]        EQUATIONS   OF  THE   FIRST  DEGREE.  105 

Let  X  denote  the  required  quantitj.     Then  from  the  condition, 

ax  -\'  h  z=:  ex  +  d^ 

d-h 

whence,        -    .    -     a;  = . 

a  —  c 

If  now  we  make  the   suppositions  that  d  =  b  and   a  zr^  c^  the 

value  of  X  becomes   -,  which   is   a   symbol    of  indeterm (nation. 

If  we   make   these   substitutions   in   the    first   equation,   it  be 
comes 

ax  +  b  =z  ax  +  b, 
an  identical  equation  (Art.  75),  which   must   be  satisfied  for    all 
values  of  x.     These   suppositions   also   render   the  conditions   of 
the  problem   so   dependent   upon   each   other,   that   any  quantity 
whatever  will  fulfil  them   all. 

Hence,  the  result  -  indicates   that  the  problem  admits  of  an 

infinite  number  of  solutions. 

3.  Find  two   quantities   such   that   a   times   the   first  increased 

by  b   times   the   second   shall   be   equal   to   c,   and  that  d  times 

the   first   increased  by  /  times   the   second   shall  be  equal  to  g. 

If  we   denote   the   quantities   by  x  and  y,  we  shall  have  from 

the   conditions  of  the   problem, 

ax-\-bi/=:c,     -     -     -     -     (1) 

dx+fy^g^     ....     (2) 

,  cd  —  aq  ,  bq  —  cf 

whence   -     y  =  — ^,      and      x  =  ^ ^,. 

bd  —  af  bd  —  aj 

If  now  we  make 

cd  =  ag,  (3)         and        afz=  bd,  (4) 
we  shall  find   by  multiplying  these  equations   together,  membei 
by  member, 

cf=bg. 

These  suppositions,  reduce  the  values  of  both  x  and  y  to  — , 

Fiom  (3)  we  find, 

^  =  ^     and  from  (4)      f=  —  xd=X 


106  ELEMENTS   OF   ALGEBRA.  [CHAP.    IV. 

which   iiubstitated  in  equation  (2),  reduce  it  to 

ax  T  hy  =:  c, 
an   equation  which   is   the   same   as   the   first. 

Under  this  supposition,  we  have  in  reality  but  one  equalicn 
between  two  unknown  quantities,  both  of  which  ought  to  be  inde- 
terminate. This  supposition  also  renders  the  conditions  of  the 
problem  so  dependent  upon  each  other,  as  to  produce  a  less 
number  of  independent  equations  than  there  are  unknowm  quan- 
tities. 

Generally,  the  result  — ,  with  the  exception  of  the  case  men- 
tioned in  Art.  71,  arises  from  some  supposition  made  upon  the 
quantities  entering  a  problem,  which  makes  one  or  more  condi- 
tions so  dependent  upon  the   others   as   to   give   rise   to   one   or 

more  indeterminate   equations.     In  these  cases  the  result   —-    is 

a  true  answer  to  the  problem,  and  is  to  be  interpreted  as 
indicating  that  the  problem  admits  of  an  infinite  number  of 
solutions. 

Inter jpretatioji   of  Negative   Results, 

89.  From  the  nature  of  tlie  signs  -\-  and  — ,  it  is  clear  that 
the  operations  which  they  indioate  are  diametrically  opposite  to 
each  other,  and  it  is  reasonable  to  infer  that  if  a  positive  re- 
sult, that  is,  one  affected  by  the  sign  +,  is  to  be  interpreted 
'n  a  certain  sense,  that  a  negative  result,  or  one  affected  by 
./he  sign  — ,  should  be  interpreted  in  exactly  the  contrary 
sense. 

To  show  that  this  inference  is  c^rect,  we  shall  discuss  one 
or  two  problems  giving  rise  to  both  positive  and  negative 
results. 

1.  To  find  a  number,  which  added  to  the  number  6,  will 
give   a    sum    equal   to   the   number   a. 

Let  X  denote  the  required  number.     Then  from,  tho  oonditions 
X  -{-h  —  a^     v  hence,      x  :=  a  —  b. 


CHAP.   IV.]         EQUATIONS   OF  THE   FIRST  DEGREE.  107 

This  farmula  will  give  the  algebraic  value  of  x  in  all  the 
particular   cases  of  the    problem. 

For  example,  let      a  =  47     and     6  =  29  ; 
then,  a;  ^47 -29  =  18. 

Again,  let  a  =  24     and     5  =  31  ; 

then,  a;  =  24  -  31  :zr  -  7. 

This  last  value  of  x^  is  called  a  negative  solution.  How  is  it 
to   be   interpreted? 

If  we  consider  it  as  a  purely  arithmetical  result,  that  is,  as 
arising  from  a  series  of  operations  in  which  all  the  quantities 
are  regarded  as  positive,  and  in  which  the  terms  add  and  sub- 
tract imply,  respectively,  augmentation  and  diminution,  the  prob- 
lem will  obviously  be  impossible  for  the  last  values  attributed 
to   a   and  b ;    for,  the   number   b  is   already   greater   than   24. 

Considered,  however,  algebraically,  it  is  not  so ;  for  we  have 
found  the  value  of  a;  to  be  —  7,  and  this  number  added,  in  the 
algebraic  sense,  to  31,  gives  24  for  the  algebraic  sum,  and  there- 
fore  satisfies   both   the   equation   and   enunciation. 

2.  A  father  has  lived  a  number  of  years  expressed  by  a;  his 
son  a  number  of  years  expressed  by  b.  Find  in  how  many  years 
the  age  of  the   son  will  be  one  fourth  the  age   of  the  father. 

Let  X   denote  the  required  number  of  years. 

Then,  a  -\-  x  will  denote  the  age  of  the  father  |  at  the  end  of  the 

and       b  -{-  X   will  denote  the  age  of  the  son      )    required  time. 

Hence,  from  the  conditions, 

— - —  =zh  -{-  x\     whence,     x  =  — — — . 
4  o 

Suppose     a  =z  54,    and    b  z=z9]     then    x  = =  —  -c:  6. 

The  father  being  54  years  old,  and  the  son  9,  in  6  yearsi  the 
father  will  be  60  years  old,  and  his  son  15 ;  now  15  is  the 
6  urth   of  60;    hence,   x  =  6   satisfies   the   enunciation. 

Let  us  now  suppose     a  =  45,    and    b  =  15  ; 

45  -  60 
then,  V  = ^ =  —  5, 


108  ELEMENTS  OF  ALGEBRA.  [CHAP.   lY. 

5   of  a;   in 


If  we   substitute   this   value   of  x   in   the   equation, 
a  -\-  X 


4 
45 5 

we  obtain, —  =  15  —  5 : 

4  ' 

or,  10  =  10. 

Hence,  —  5  substituted  for  x^  verifies  the  equation,  and  there- 
fore  is  a  true  answer. 

Now,  the  positive  result  which  was  obtained,  shows  that  the 
age  of  the  father  will  be  four  times  that  of  the  son  at  the 
expiration  of  6  years  from  the  time  when  their  ages  were 
considered ;  while  the  negative  result,  indicates  that  the  age  of 
the  father  was  four  times  that  of  his  son,  5  years  previous  to 
the   time  when   their   ages   were   compared. 

The  question,  taken  in  its  general,  or  algebraic  sense,  demands 
the  time,  when  the  age  of  the  father  is  four  times  that  of  the 
son.  In  stating  it,  we  supposed  that  the  time  was  yet  to 
come ;  and  so  it  was  by  the  first  supposition.  But  the  con- 
ditions imposed  by  the  second  supposition,  required  that  the 
time  should  have  already  passed,  and  the  algebraic  result  con- 
formed to   this   condition,  by  appearing  with   a  negative   sign. 

Had  we  wished  the  result,  under  the  second  supposition,  to 
have  a  positive  sign,  we  might  have  altered  the  enunciation 
by  demanding,  how  many  years  since  the  age  of  the  father  loas 
four  times  that  of  the  son. 

If  X  denote  the  number  of  years,  we  shall  have  from  the 
conditions, 

a  —  X       .  *  4b  —  a 

— - —  =  b  —  x:     hence,     x  = — - — . 

4  '  3 

If  a  =  45     and     b  =  15,     x  will  be  equal  to  5. 

From  a  careful  consideration  of  the  preceding  discussion,  we 
may  deduce  the  following  principles  with  regard  to  negative 
results. 

1st.  Every  negative  value  found  for  the  unknown  quft^ntiiy  from 
an  equation  of  the  first  degree,  will,  when  taken  with  ^^  proper 
sign^  satisfy  the  equation  from  which  it  was  derived. 


CHAP    lY.]         EQUATIONS  OF  THE   FIRST  DEGREE.  109 

2d.  This  negative  value,  taken  with  its  proper  sig?^,  will  also 
satisfy,  the  conditions  of  the  problem,  unrJerstood  in  its  algebraic 
sense, 

Sd.  If  a  positive  result  is  interpreted  in  a  certain  sense,  a  nega- 
tive result  must  be  interpreted  in  a  directly  contrary  sense, 

4th.  The  negative  result,  with  its  sign  changed,  may  be  regarded 
as  the  answer  to  a  problem  of  which  the  enunciation  only  differs 
from  the  one  proposed  in  this :  that  certain  quantities  which  were 
additive   have  become   subtractive,   and  the   reverse, 

90«  As  a  further  illustration  of  the  extent  and  power  of  the 
algebraic  language,  let  us  resume  the  general  problem  of  the 
laborer,  already  considered. 

Under   the   supposition  that  the  laborer  receives  a  sum  c,  we 

have  the   equations 

X  -\-    y  z=i  n)         .  bn  -{-  c  an  —  c 

y      whence,     x  i= —7-,     y  = -—^, 

ax  —  by  z=z  c  )  a-\-  b^  a  +  0 

If,  at  the  end  of  the  time,  the  laborer,  instead  of  receiving 
a  sum  c,  owed  for  his  board  a  sum  equal  to  c,  then,  by  would 
be  greater  than  ax,  and  under  this  supposition,  we  should  have 
the  equations, 

x  -\-  y  =:n,     and     ax  —  by  =.  —  c. 

Now,  since  the  last  two  equations  differ  from  the  preceding 
two  given  equations  only  in  the  sign  of  c,  if  we  change  the 
sign  of  c,  in  the  values  of  x  and  y,  found  from  these  equations, 
the  results  will  be  the  values  of  x  and  y,  in  the  last  equa- 
tions :  this  gives 

_  bn  —  c  _  an  -\-  c 

The  results,  for  both  enunciations,  may  be  comprehended  in 
the   same  formulas,  by  writing 

^  bndbc  ^  an  zjp  c 

^~    a  +  b'       ^  ~    a  +  b' 

The  double  sign   it,  is  read  plus  or   minus,  and  qp,  is  read, 

minus    or  plus.      The    upper    signs    correspond   to   the   case  in 

which  the  laborer  received,  and  the  lower  signs,  to   the  case  in 


110  ELEMENTS  OF  ALGEBRA^  [CHAP.   IV. 

which  he  owed  a  sum  c.  These  formulas  also  comprehend  the 
case  in  which,  in  a  settlement  between  the  laborer  and  hid 
employer,  their  accounts  balance.  This  supposes  c  =  0,  which 
gives 

_^     hn  ^    an 

Discussion  of  Problems, 

91.  The  discussion  of  a  problem  consists  in  making  e^ery 
possible  supposition  upon  the  arbitrary  quantities  which  enter 
the  equation  of  the  problem,  and  interpreting  tne   results. 

An  arbitrary  quantity,  is  one  to  which  we  may  assign  a  value^ 
at  pleasure. 

In  every  general  problem  there  is  always  one  or  more  arbi 
trary  quantities,  and  it  is  by  assigning  particular  values  to 
these   that  we  get  the  particular   cases  of  the  general  problem. 

The  discussion  of* the  following  problem  presents  nearly  all 
the  circumstances  which  are  met  with  in  problems  giving  rise 
to    equations  of  the  first   degree. 

PROBLEM    OF    THE    COUJIIERS. 

Two  couriers  are  traveling  along  the  same  right  line  and 
in  the  same  direction  from  R'  toward  R.  The  number  of  miles 
traveled  by  one  of  them  per  hour  is  expressed  by  m,  and  the 
number  of  miles  traveled  by  the  other  per  hour,  is  expressed 
by  n,  Noy,  at  a  given  time,  say  12  o'clock,  the  distance  be- 
tween them  is  equal  to  a  number  of  miles  expressed  by  a :  re- 
quired  the   time  when   they  are   together. 

R^ __A B R^ 

At  12  o'clock,  suppose  the  forward  courier  to  be  at  B,  the 
other  at  A,  and  R  or  R'  to  be  the  point  at  which  they  are 
together. 

Let  a  denote  the  distance  AB,  between  the  couriers  at  12 
o'clock,  and  suppose  that  distances  measured  to  the  right,  from 
A,  are   regarded   as  positive   quantities. 


CHAP.   IV.J       EQUATIONS  OF  THE   FIRST  DEGREE.  Ill 

Let  t  denote  the  number  of  hours  frcm  12  o'clock  to  the 
time   when   they   are   together. 

Let  X  denote  the  distance  traveled  by  the  forward  courier 
in   t  hours ; 

Then,  a-\-  x  will  denote  the  distance  traveI<Kl  by  the  other 
in  the   same   time. 

Now,  since   the  rate  per  hour,  multiplied  by  the  number  of 
hours,  gives   the   distance   passed   over  by  each,  we   have, 
t  X  m  z=za  -{-  X     -    -     -     -     (1) 
t  X  n  =x  -     -     -     -     (2). 

Subtracting   the   second  equation  from  the  first,  member  from 

member,  we  have, 

t{m  —  n)  =  a', 

whence,     -    -    -    -     t  = . 

m  —  n 

We  will  now  discuss  the  value  of  ^ ;  a,  m  and  7^,  bein^ 
arbitrary  quantities. 

Mrst,   let  us    suppose   m  >  w. 

The  denominator  in  the  value  of  t,  is  then  positive,  and  since 
a  is  a  positive  quantity,  the   value  of  t  is   also   positive. 

This  result  is  interpreted  as  indicating  that  the  time  when 
they  are  together  is  after   12  o'clock. 

The    conditions    of    the    problem    confirm  this    interpretation. 

For  if  m^  n^  the  courier  from  A  will  travel  faster  than  the 
courier  from  B,  and  will  therefore  be  continually  gaining  on 
him :  the  interval  which  separates  them  will  diminish  more  and 
more,  until  it  becomes  0,  and  then  the  couriers  will  be  found 
upon  the  same  point  of  the  line. 

In  this  case,  the  time  ^,  which  elapses,  must  be  added  to  12 
o'clock,  to   obtain  the  time  when  they   are   together. 

Second^   suppose   m  <^n. 

The  denominator,  m  —  n  will  then  be  negative,  and  the  value 
of  t  will   also   be   negative. 

This  result  is  interpreted  in  a  sense  exactly  contrary  to  the 
interpretation  of  the  positive  result ;  that  is,  it  indicates  that 
the  time  of  their   being  together  was  previous  to   12   o'clock. 


112  ELEMENTS  OF  ALGEBRA.  [CHAP.   IV. 

This  interpretation  is  also  confirmed  by  considering  the 
circumstances  of  the  problem.  For,  under  the  second  suppo- 
Bition,  the  courier  which  is  in  advance  travels  the  fastest,  and 
therefore  will  continue  to  separate  himself  from  the  other 
oourier.  At  12  o'clock  the  distance  between  them  was  equal 
bo  a\  after  12  o'clock  it  is  greater  than  a;  and  as  the  rate 
of  travel  has  not  been  changed,  it  follows  that  previous  to  12 
o'clock  the  distance  ^must  have  been  less  than  a.  At  a  certain 
hour,  therefore,  before  12,  the  distance  between  them  must  have 
b^en  equal  to  nothing,  or  the  couriers  were  together  at  some 
point  R'.  The  precise  hour  is  found  by  subtracting  the  value  of 
t  from   12   o'clock. 

Third,    suppose   m  =z  n. 

The  denominator  m  —  n   will   then   become   0,    and  the  value 

of  i  will  reduce  to  -,  or  oo . 

This  result  indicates  that  the  length  of  time  that  must  elapse 
before  they  are  together  is  greater  than  any  assignable  time,  or 
In  other  words,  that  they  will  never  be  together. 

This  interpretation  is  also  confirmed  by  the  conditions  of  the 
problem. 

For,  at  12  o'clock  they  are  separated  by  a  distance  a,  and  if 
m  =  n  they  must  travel  at  the  same  rate,  and  we  see,  at  once, 
that  whatever  time  we  allow,  they  can  never  come  together; 
hence,  the  time  that  must  elapse  is  infinite. 

Fourth^   suppose  a  =  0   and   m^  n   or   m<^n. 

The  numerator  being  0,  the  value  of  the  fraction  is  0  oi 
<  =  0. 

This  result  indicates  that  they  are  together  at  12  o'clock, 
or  that  there  is  no  time  to  be  added  to  or  subtracted  fi^om 
12  o'clock. 

The  conditions  of  the  problem  confirm  this  interpretation. 
Because,  if  a  =  0,  the  couriers  are  together  at  12  o'clock ;  and 
since  they  travel  at  different  rates,  they  could  never  have  been 
together,  nor  can  they  be  together  after  12  o'clock:  hence,  t  can 
have  no  other   value  than   0. 


4m AP.   IV.]  OF   INEQUALITIES.  11 J 

Fifths  suppose,  a  =  0   and  m  =n. 

The   value  of  i  becomes   -,  an   indeterminate  result. 

This  indicates  that  t  may  have  any  value  whatever,  or  in 
other  words,  that  the  couriers  are  together  at  any  time  either 
before  or  after  12  o'clock :  and  this  too  is  evident  from  the  cir 
cumstances  of  the  problem. 

For,  if  a  ==  0,  the  couriers  are  together  at  12  o'clock ;  and 
since  they  travel  at  the  same  rate,  they  will  always  be  together; 
hence,  t  ought  to  be  indeterminate. 

The    distances   traveled   by   the   couriers    in    the    time    t  are, 

respectively, 

ma  _       na 

and    , 

m  —  n  m^n 

both  of  which  will  be  plus  when  m^n,  both  minus  when  m  <  w, 
and  infinite  when  m  z=  n. 

In  the  first  case  t  is  positive ;  in  the  second,  negative;  and  in 
the   third,  infinite. 

When  the  couriers  are  together  before  12  o'clock,  the  distances 
are  negative,  as  they  should  be,  since  we  have  agreed  to  call 
distances  estimated  to  the  right  positive^  and  from  the  rule  for 
interpreting  negative  results,  distances  to  the  left  ought  to  be 
regarded  as  negative. 

Of  Inequalities, 

92 1  An  inequality  is  the  expression  of  two  unequal  quantities 
connected  by   the  sign  of  inequality. 

Thus,  a  >  i  is  an  inequality,  expressing  that  the  quantity  a 
is  greater  than  the  quantity   b. 

The  part  on  the  left  of  the  sign  of  inequality  is  called  the  first 
member,  that  on  the  right  the  second  member. 

The  operations  which  may  be  performed  upon  equations,  may 
in  general  be  performed  upon  inequalities;  but  there  are,  never- 
theless, some  exceptions. 

In  order  to  be  clearly  understood,  we  will  give  examples  of 
the  different  transformations  to  which  ir  equalities  may  lb©  sub 

8 


114  ELEMENTS  OF  ALGEBRA.  [CHAP.   IV. 

• 

jected,  taking  care  to   point  out   the   exceptions   to  which   these 
transformations   are   liable. 

Two  inequalities  are  said  to  subsist  in  the  same  sense,  when 
the  greater  quantity  is  in  the  first  member  in  both,  or  in  the 
second  member  in  both;  and  in  a  contrary  sense,  when  the 
greater  quantity  is  in  the  first  member  of  one  and  in  the  second 
member  of  the  other. 

Thus,     25  >  20     and     18  >  10,     or    6  <  8     and    7  <  9, 
are   inequalities   which   subsist  in   the   same   sense;    and   the   in 
equalities 

15  >  13     and     12  <  14, 
subsist  in  a  contrary  sense. 

\,  If  we  add  the  same  quantity  to  both  members  of  an  inequality^ 
or  subtract  the  same  quantity  from  both  member s^  the  resulting 
inequality  will  subsist  in  the  same  sense. 

Thus,   take    8  >  6 ;   by   adding   5,  we   still  have 
b-i-5>6  +  5; 
and   subtracting   5,  we  have 

8  -  5  >  6  -  5. 

When  the  two  members  of  an  inequality  are  both  negative, 
that  one  is  the  least,  algebraically  considered,  which  contains  the 
greatest  number  of  units. 

Thus,  —  25  <  —  20 ;  and  if  30  be  added  to  both  members, 
we  have  5  <  10.  This  must  be  understood  entirely  in  an  alge- 
braic sense,  and  arises  from  the  convention  before  established,  to 
consider  all  quantities  preceded  by  the  minus  sign,  as  subtractive. 

The  principle  first  enunciated  serves  to  transpose  certain  terms 
from  one  member  of  the  inequality  to  the  other.  Take*  for  ex 
ample,  the  inequality 

a2  +  62>352-_2a2; 

there  will  result,  by  transposing, 

a2  -f  2a2  >  352  —  b^,     or     Sa^  >  262. 

2.  If  two  inequalities  subsist  in  the  same  sense^  and  we  add  them 
member  to  member^  the  resulting  inequality  will  also  subsist  in  the 
same 


CHAP.  IV.J  OF  INEQUALITIES.  115 

Thus,  if  we  add  a  >  5  and  c  >  c?,  member  to  member, 
there  results  a  +  c"^  b  +  d. 

But  this  is  not  always  the  case,  when  we  subtract,  member  from 
member,  two  inequalities  established  in  the  same  sense. 

Let  there  be   two   inequalities    4  <  7    and    2  <  3,   we    have 

4-2    or    2<7~3     or    4. 
But  if  we  have  the  inequalities  9  <  10    and    6  <  8,   by   sub- 
tracting,  we   have 

9-6     or     3  >  10  —  8     or    2. 

We  should  then  avoid  this  transformation  as  much  as  possible, 
or  if  we  employ  it,  determine  in  which  sense  the  resulting  ia- 
equality  subsists. 

3.  If  the  two  members  of  an  inequality  be  multiplied  by  a 
positive  quantity,  the  resulting  inequality  will  exist  in  the  same 
sense. 

Thus,     -    -     -     cL<^b,    will   give    3a  <  35 ; 

and,      -    -    -    -     —  a  <  —  5,     —  3a  <  —  36. 

This  principle  serves  to  make  the  denominators  disappear. 

_  ,      .  ,.  d?  —  b'^       (?  —  d^ 

From  the  mequality  — — —  >  — , 

AiCL  oa 

we  deduce,  by  multiplying  by   ^ad, 

3a(a2_52)>2c^(c2-(^2), 

and  the  same  principle  is  true  for  division.     But, 

When  the  two  members  of  an  inequality  are  multiplied  or 
divided  by  a  negative  quantity,  the  resulting  inequality  will  sub- 
tlst   in   a  contrary   sense. 

Take,  for  example,     8  >7;  multiplying  by    —3,  we  have 

-24<  -21. 

8  8  7 

In  like  manner,   8  >  7  gives  — — ,     or ^  <  "~  T- 

—  o  o  o 

Therefore,  when  the  two  members  of  an  inequality  are  multi- 
plied  or  divided  by  a  quantity,  it  is  necessary  to  ascertain 
whether  the  multiplier  or  divisor  is  negative;  for,  in  that  case, 
the  inequality  will  exist  in  a  contrary  sense. 


116  ELEMENTS  OF  ALGEBRA.  [CHAP.   IV 

4.  It  is  not  permitted  to  change  the  signs  of  the  two  members 
of  an  inequality^  unless  we  establish  the  resulting  inequality  in  a 
contrary  sense;  for,  this  transformation  is  evidently  the  same  as 
multiplying  the  two  members  by    —  1. 

5.  Both  members  of  an  inequality  between  positive  quantities 
tan  be  squared^  and   the  inequality   will  exist  in   the  same   sense. 

Thus,   from   5  >  3,   we  deduce,   25  >  9  ;   from  a  +  5  >  c,  we 

find  I 

{a  +  by  >  c\ 

6.  When  the  signs  of  both  members  of  the  inequality  are  not 
known  J  we  cannot  tell  before  the  operation  is  performed,  in  which 
sense   the   resulting   inequality  will  exist. 

For  example,    —  2  <  3    gives    {  —  2)^    or    4  <  9. 
But,     3  >  —  5    gives,  on  the  contrary,   (3)^    or    9  <  (  —5)^ 
or   25. 

We  must,  then,  before  squaring,  ascertain  the  signs  of  the  two 
members. 

Let  us  apply  these  principles  to  the  solution  of  the  following 
examples.  By  the  solution  of  an  inequality  is  meant  the  oper 
ation  of  finding  an  inequality,  one  member  of  which  is  the 
unknown  quantity,  and  the  other  a  known  expression. 

EXAMPLES. 

1.  5a;-6>19.  Ans.    a;  >  5. 

14 

2.  3a;  +  -—  a;  —  30  >  10.  Ans.    a:  >  4, 

4.  -—  +  5a;  —  a5  >  — -  Ans,    x^cu    ' 

5  5 

5.  -= —  aa;  +  a5  <  — .  Ans.    jc  <  i. 


CHAPTER  V. 

EXTRACTION  OF  THE  SQUARE  ROOT  OF  NUMBERS.'-  ORMATION  OP  THB 
SQUARE  AND  EXTRACTION  OF  THE  SQUARE  ROOT  OF  ALGEBRAIC  QUANTI- 
TIES.  TRANSFORMATION  OF  RADICALS  OF  THE  SECOND  DEGREE. 

93  •  The  square  or  second  power  of  a  number,  is  the  product 
which  arises  from  multiplying  that  number  by  itself  once :  for 
example,  49  is   the   square  of  7,  and  144  is   the  square  of  12. 

The  square  root  of  a  number,  is  that  number  which  multiplied 
by  itself  once  will  produce  the  given  number.  Thus,  7  is  the 
square  root  of  49,  and  12  the  square  root  of  144 :  for,  7x7  =  49, 
and   12  X  12  =  144. 

The  square  of  a  number,  either  entire  or  fractional,  is  easily 
found,  being  always  obtained  by  multiplying  the  number  by  itself 
once.  The  extraction  of  the  square  root  is,  however,  attended 
with  some  difficulty,  and  requires  particular   explanation. 

The  first  ten  numbers  are, 

1,    2,     3,      4,       5,       6,      7,      8,       9,       10, 
and  their   squares, 

1,     4,     9,     16,     25,     86,    49,     64,     81,     100. 

Conversely,  the  numbers  in  the  first  line,  are  the  square  rooti 
of  the   corresponding  numbers   in   the   second   line. 

We  see  that  the  square  of  any  number,  expressed  by  one 
figure,  will  contain  no   unit  of  a  higher  order  than  tens. 

The  numbers  in  the  second  line  are  jperfect  squares.,  and, 
generally,  any  number  which  results  from  multiplying  a  whole 
number  by  itself  once,  is  a  perfect  square. 

If  we  wish  to  find  the  square  root  of  any  number  less,  than 
100,  we  look  in  the  second  line,  above  given,  and  if  the  num- 
ber is  there  written,  the  corresponding  number  in   ?hQ  first  line 


118  ELEMENTS  OF  ALGEBRi.  fCHAP.   V. 

is  its  square  root.  If  the  number  falls  between  any  two  num 
bers  in  the  second  line,  its  square  root  will  fall  between  the 
corresponding  numbers  in  the  first  line.  Thus,  55  falls  between 
49  and  64 ;  hence,  its  square  root  is  greater  than  7  and  less 
than  8.  Also,  91  falls  between  81  and  100;  hence,  its  square 
root  is   greater   than   9   and  less   than   10. 

If  now,  we  change  the  units  of  the  first  line,  1,  2,  3,  4,  &c., 
into  units  of  the  second  order,  or  tens,  by  annexing  0  to  each, 
we  shall  have, 

10,  20,  30,  40,  50,  60,  70,  80,  90,  100, 
and  their   corresponding  squares  will  be, 

100,  400,  900,  1600,  2500,  3600,  4900,  6400,  8100,  10000: 
Hence,  the  square  of  any  number  of  tens  will  contain  no  unit  of 
u    less   denomination    than   hundreds, 

94.  We  may  regard  every  number  as  composed  of  the  sum 
of  its   tens  and   units. 

J^^ow,  if  we  represent  any  number  by  N"^  and  denote  the 
tens  by  a,  and   the  units  by   6,  we   shall   have, 

whence,  by  squaring   both   members, 

iV^2  _  ^2  _|_  2ah  +  52  : 

Hence,  the  square  of  a  number  is  equal  to  the  square  of  the 
tens,  plus  twice  the  product  of  the  tens  by  the  units,  plus  the  square 
of  the   units. 

For   example,  78  =  70  +  8,     hence, 

(78)2  =  (70)2  +  2  X  70  X  8  +  (8)2  =  4900  +  1120  +  64  =••  6084. 

95.  Let  us  now  find   the   square   root   of  6084. 
Since  this  number  is  expressed  by  more  than  two 

figures,  its  root  will  be  expressed  by  more  than  one.  60  84 

But  since  it  is   less  than  10000,  which  is  the  square 
of  100,  the   root  will   contain   but  two   places   of  figures ;    that 
is,  i:mts  and  tens.  * 

Now,  the  square  of  the  number  of  tens  must  be  found  in  the 
number  expressed  by  the  two  left-hand  figures,  which  we  will 
separate  from  the   other  two,  by  placing  a  point  over  the  place 


60  84 1 78 
49 

7  X  2  =  14  8 


1184 
1184 
0 


CHAP,   v.]  SQUARH  ROOT  OF  NUMBERS.  119 

of  units,  and  another  over  the  place  of  hundreds.  These  parts, 
of  two  figures  each,  are  called  periods.  The  part  60  is  com- 
prised between  the  two  squares  49  and  64,  of  which  the  roots 
are  7  and  8  :  hence,  7  is  the  number  of  tens  sought ;  and  the 
required  root  is  composed  of  7  tens  plus  a  certain  number  of 
units. 

The  number  7  being  found,  we 
set  it  on  the  right  of  the  given 
number,  from  which  we  separate 
it  by  a  vertical  line :  then  we 
subtract  its  square  49  from  60, 
which  leaves  a  remainder  of  11, 
to  which  we  bring  down  the  two 
next  figures  84.  The  result  of  this  operation  is  1184,  and  this 
number  is  made  up  of  twice  the  product  of  the  tens  by  the  units 
plus  the  square  of  the  units. 

But  since  tens  multiplied  by  units  cannot  give  a  product  of  a 
lower  order  than  tens,  it  follows  that  the  last  number  4  can 
form  no  part  of  double  the  product  of  the  tens  by  the  units : 
this  double  product,   is,   therefore  found  in   the  part   118. 

Now,  if  we  double  the  number  of  tens,  which  gives  14,  and 
then  divide  118  by  14,  tb.  quotient  8  is  the  number  of  units  (f 
the  root,  or  a  greater  number.  This  quotient  can  never  be  too 
small,  since  the  part  118  will  be  at  least  equal  to  twice  the 
product  of  the  number  of  tens  by  the  units:  but  it  may  be  too 
large;  for  the  118,  besides  the  double  product  of  the  number 
of  tens  by  the  units,  may  likewise  contain  tens  arising  from 
the   square  of  the   units. 

To  ascertain  if  the  quotient  8  expresses  the  number  of  units, 
we  place  the  8  to  the  right  of  the  14,  which  gives  148,  and  then 
we   multiply  148  by  8 :    Thus,  we  evidently  form, 
1st,  the  square  of  the  units ;  and 
2d,   the  double  product  of  the  lens  by  the  units. 
This  multiplication  being  affected,    gives   for   a   product    1184, 
tiift  same  number  as   the  result  of  the  first   operation.      Having 


120  ELEMENTS  OF  ALGEBR4.  [CHAP.  V, 

subtracted  ".he  product,  we  find  the  remainder  equal  to  0 :  hence 
78,  is   the   root   required. 

Indeed,  in  the  operations,  we  have  merely  subtracted  from  the 
given  number  6084, 

1st,  the  square  of  7   tens  or  of  70 ; 

2d,  twice  the  product  of  70  by  8;   and 

3d,  the  square  of  8 :  that  is,  the  three  parts  which  enter  mto 
tbe  composition  of  the  square  of  78. 

In  the  same  manner  we  may  extract  the  square  root  of  any 
number  expressed  by  four  figures. 

95.  Let  us  now  extract  the  square  root  of  a  number  expressed 
by  more  than  four  figures. 

Let  56821444  be  the  number.  56  82  14  44  |  7538 

If   we   consider   the    root    as    the  49  • 

sum   of    a   certain    number   of   tens  14  5  78  2 

and   a  certain   number  of  units,  the  725 

given    number    will,    as    before,    be  150  3  57i  4 

oqual  to  the  square  of  the  tens  plus  450  9 

twice   the    product   of   the   tens    by  150^8^ 


12054  4 
12054  4 


the  units  plus  the  square  of  the  units. 

If  then,  as  before,  we  point  off 
a  period  of  two  figures,  at  the  right,  the  square  of  the  tens  of  the 
required  root  will  be  found  in  the  number  568214,  at  the  left ; 
and  the  square  root  of  the  greatest  perfect  square  in  this  number 
will  express   the  tens   of  the  root. 

But  since  this  number,  568214,  contains  more  than  two  figures, 
its  root  will  contain  more  than  one,  (or  hundreds),  and  the  ^qaare 
of  the  hundreds  will  be  found  in  the  figures  5682,  at  the  left  of  14  ; 
hence,  if  we  poi«it  off  a  second  period  14,  the  square  root  of  the 
greatest  perfect  square  in  5682  will  be  the  hundreds  of  the  required 
root.  But  since  5682  contains  more  than  two  figures,  its  root  will 
contain  more  than  one,  (or  thousands),  and  the  square  of  the  thousands 
will  be  found  in  56,  at  the  left  of  82 :  hence,  if  we  point  off  a  third 
period  82,  the  square  root  of  the  greatest  perfect  square  in  56  will 
be  the  thousands  of  the  required  root.  Hence,  we  place  a  point 
over  56.  and  then  proceed  thus  : 


CHAP.   V.J  SQUARE  ROOT  OF  NUMBERS.  121 

Placing  7  on  the  right  of  the  given  number,  and  subtracting 
its  square,  49,  from  the  left  hand  period,  we  find  7  for  a  remain- 
der, to  which  we  annex  the  next  period,  82.  Separating  the  last 
figure  at  the  right  from  the  others  by  a  point,  and  dividing  the 
number  at  the  left  by  twice  7,  or  14,  we  have  5  for  a  quotient 
figure,  which  we  place  at  the  right  of  the  figure  already  found, 
and  also  annex  it  to  14.  Multiplying  145  by  5,  and  subtracting 
the  product  from  782,  we  find  the  remainder  57.  Hence,  75  is 
the  number  of  tens  of  tens,  or  hundreds,  of  the  required  square 
root. 

To  find  the  number  of  tens,  bring  down  the  next  period  and 
annex  it  to  the  second  remainder,  giving  5714,  and  divide"  571 
by  double  75,  or  by  150.  The  quotient  3  annexed  to  75  gives 
753  for  the  number  of  tens  in  the  root  sought. 

We  may,  as  before,  find  the  number  of  units,  which  in  this 
case  will  be  8.  Therefore,  the  required  square  root  is  7538.  A 
similar  course  of  reasoning  may  be  applied  to  a  number  expressed 
by  any  number  of  figures.  Hence,  for  the  extraction  of  the 
square  root  of  numbers,  we  have  the  following 

RULE. 

I.  Separate  the  given  number  into  periods  of  two  figures  eack^ 
beginning  at  the  right  hand:  the  period  on  the  left  will  often  con- 
tain but  one  figure, 

n.  Find  the  greatest  perfect  square  in  the  first  period  on  the 
left,  and  place  its  root  on  the  right  after  the  manner  of  a  quotient 
in  division,  S^ibtract  the  square  of  this  root  from  the  first 
period,  and  •  to   the  remainder  bring  down  the  second  period  for  a 


III.  Double  the  root  already  found  and  place  it  on  the  left  for  a 
divisor.  See  how  many  times  the  divisor  is  contained  in  the 
dividend,  exclusive  of  the  right  hand  figure,  and  place  the  quotien 
in  the  root  and  also  at  the  right  of  the  divisor, 

IV,  Multiply  the  divisor,  thus  augmented,  by  the  last  figure 
of  the  root  found^   and   subtract    the  product  from   the   dividend 


122  ELEMENTS  OF  ALGEBR^.  I  CHAP.  V, 

and  to   the    remainder    bring   down   the  next   period  for  a   new 
dividend, 

V.  Double  the  whole  root  already  found^  for  a  new  divisor, 
and  continue  the  operation  as  before,  until  all  the  periods  are 
brought  down. 

Remark  I. — If, ,  after  all  the  periods  are  brought  down,  there 
is  no  remainder,  the  proposed  number  is  a  perfect  square.  But 
if  there  is  a  remainder,  we  have  only  found  the  root  of  the 
greatest  perfect  square  contained  in  the  given  number,  or  the 
entire  part   of  the   root  sought. 

For  example,  if  it  were  required  to  extract  the  square  root  of 
168,  we  should  find  12  for  the  entire  part  of  the  root  and  a 
remainder  of  24,  which  shows  that  168  is  not  a  perfect  square. 
But  is  the  square  of  12  the  greatest  perfect  square  contained 
in   168?     That  is,  is    12   the   entire  part  of  the  roof? 

To  prove  this,  we  will  first  show  that,  the  difference  between 
the  squares  of  two  consecutive  numbers,  is  equal  to  twice  the  less 
number  augmented  by  1. 

Let  a    represent  the  less  number, 

and  «  -f-  1,    the  greater. 

Tlien,  (a  +  If  =  a^-Y2a  +  \, 
and  {ay  z=  a^, 


their  difference  is  2a  +  1     as   enunciated  :   hence, 

The  entire  part  of  the  root  cannot  be  augmented  by  1,  unless 
the  remainder  is  equal  to,  or  exceeds  twice  the  root  found,  plus  1. 

But,  12x2+1=  25;  and  since  the  remainder  24  is  less 
than  25,  it  follows  I'nat  12  cannot  be  augmented  by  a  number 
as   great  as   unity  :  hence,  it  is   the   entire  part   of  the   root. 

The  principle  demonstrated  above,  may  be  readily  applied  in 
finding   the   squares   of  consecutive  numbers. 

If  the  numbers  are  large,  it  will  be  much  easier  to  apply  the 
above  pifnciple  than  to  square   the  numbers  separately. 


CHAP,  v.]  SQUARE  ROOT  OF  NUMBERS.  128 

For  example,  if  we  have     (651)2  ^  423801, 
and  wish  to  find  the  square  of  652,  we  have, 
(651)2  ^  423801 
-h  2  X  651     =      1302 
+  1     =  1 


and 

(652)2  ^  425104. 

Also, 

(652)2  ^  425104 

+  2  X  652  =      1304 

+  1  =            1 

and 

(653)2  ^  426409. 

Remark  II. — The  number  of  places  of  figures  in  the  root 
will  always  be  equal  to  the  number  of  periods  into  which  the 
given  number  is  separated. 

EXAMPLES. 

1.  Find  the   square  root  of  7225. 

2.  Find   the   square  root  of  17689. 

3.  Find   the   square  root  of  994009. 

4.  Find  the   square  root  of  85678973. 

5.  Find  the   square  root  of  67812675. 

6.  Find  the   square  root  of  2792401. 

7.  Find   the   square  root  of  37496042. 

8.  Find  the  square  root  of  3661097049. 

9.  Find  the  square  root  of  918741672704. 

Remark  III. — The  square  root  of  an  imperfect  square,  is  in 
commensurable  with  1,  that  is,  its  value  cannot  be  expressed 
in   exact  parts   of  1. 

To  prove  this,  we  shall  first  show  that  if    -=-     is  an  irreduci- 

b 

ble  fraction,  its  square     -7^     must  also  be  an  irreducible  fraction. 

A  number  is  said  to  be  prime  when  it  cannot  be  exactly  di- 
vided by  any  other  number,  except  1.  Thus  3,  5  and  7  are 
prime  numbers. 


124  ELEMENTS   OF  ALGEBRA.  [CHAP.   V. 

It  is  a  fundamental  principle,  that  every  number  may  be  re^ 
solved  into  prime  factors,  and  that  any  number  thus  resolved, 
is  equal  to  the  continued  product  of  all  its  prime  factors.  It 
often  happens  that  some  of  these  factors  are  equal  to  each 
other.     For   example,  the  number 

50  =  2  X  5  X  5 ;       and,      180  =  2  X  2  X  3  X  3  x  5. 

Now,  from  the  rules  for  multiplication,  it  is  evident  that  the 
square  of  any  number  is  equal  to  the  continued  product  of  all 
the  prime  factors  of  that  number,  each  taken  twice.  Hence,  we 
see  that,  the  square  of  a  number  cannot  contain  any  prime  factor 
which  is  not  contained  in  the  number  itself 

But,  since     — ,     is,   by   hypothesis,  an  irreducible   fraction,   a 

and    b    can    have   no   common   factor :    hence,    it   follows,    from 
what  has  just  been   shown,  that   a^   and   b^   cannot  have  a  com- 


a 


2 


mon  factor,  that  is,    —     is    an    irreducible    fraction,    which   was 
to   be   proved. 


rvS 


For  like  reasons,  -r-,     tt,   -   -  -;—•>  are  also  irreducible  fractions. 

Now,  let  c  represent  any  whole  number  which  is  an  imper 
feet  square.  If  the  square  root  of  c  can  be  expressed  by  » 
fraction,   we   shall  have 

in  which   -—    is  an  irreducible  fraction. 
6 

Squaring   both   members,  gives, 

''■-¥^ 

or  a  whole  number   equal   to   an   irreducible   fraction,    which   is . 
absurd ;   hence,  ,^/c"  cannot  be   expressed  by   a  fraction. 

We  conclude,  therefore,  that  the  square  root  of  an  imperfect 
square  cannot  be  expressed  in  exact  parts  jf  1.  It  may  be 
shown,  in  a  similar  manner,  that  any  root  of  an  irnperfeci 
power  of  the  decree  indicated,  cannot  be  expressed  in  exact  parts 
of  1. 


CHAP,   v.]  SQUARE   ROOT  OF  FRACTIONS.  125 

Extraction  of  the  Square  Root  of  Fractions. 

96.  Since  the  second  power  of  a  fraction  is  obtained  by 
squaring  tha  numerator  and  denominator  separately,  it  follows 
that  the  square  root  of  a  fraction  will  be  equal  to  the  square  root 
of  the  numerator  divided  by  the  square  root  of  the  denominator. 

For  example,  y -^  =  y, 

a         a        a? 
smce  "T  ><  "F  =  IT- 

0  0  0^ 

But  if  the  numerator  and  the  denominator  are  not  both  per- 
fect squares,  the  root  of  the  fraction  cannot  be  exactly  found. 
We  can,  however,  easily  find  the  root  to  within  less  than  the 
fractional  unit. 

Thus,  if  we  were  required  to   extract  the  square  root  of  the 

fraction   -7-,  to  within  less  than  — ,  multiply  both  terms  of  the 

fractions  by  6,  and  we  have  — . 

Let  r^  represent  the  greatest  perfect  square  in  a5,  then  will 
ah  be  contained  between  r^  and  (r  +  1)^,  and  —  will  be  con- 
tained  between 

and   the  true   square  root  of  ttt  =  -7-?   will    be    contained  be- 

0^         0 

kween 

T  T  "4-  1  1. 

but  the  difference  between  -r-  and  — ; —    is   -7-;  hence,  either 

o  0  0 

I 

will  be  the  square  root  of  -r-j  to  within  less  than  -y-.     We  Vave 

0  0 

then  the  folio  wiiLg 


126  ELEMENTS  OF  ALGEBR4.  LCHAP.   V. 

KULE. 

Multiply  the  numerator  hy  the  denominator^  and  extract  the 
square  root  of  the  product  to  within  less  than  1  ;  divide  the 
result  by  the  denominator,  and  the  quotient  will  be  the  approxi- 
mate  root. 

For  example,   to   extract  the  square  root  of  —,  we  multiply 

o 

3  by  5,   which  gives   15 ;   the  perfect  square  nearest  15,  is  16, 

4  3 

and  its  square  root  is  4 ;  hence,  — -  is  the  square  root  of  -— 

O  u 

to   within   less   than   — -. 
5 

97 1  If   we    wish    to   determine   the   square   root    of    a  whole 

number   which   is    an    imperfect   square,   to  within    less   than   a 

given  fractional  unit,  as  —,  for  example,  we  have  only  to  place 

the  number  under  a  fractional  form,  having  the  given  fractional 
unit  (Art.  63),  and  then  we  may  apply  the  preceding  rule:  or 
what  is  an  equivalent  operation,  we  may 

Multiply  the  given  number  by  the  square  of  the  denominator 
of  the  fraction  which  determines  the  degree  of  approximation ;  then 
extract  the  square,  root  of  the  product  to  the  nearest  unit,  and 
divide    this   root   by    the   denominator  of  the  fraction. 

EXAMPLES. 

1.  Let  it  be  required  to  extract  the  square  root  of  59,  to 
within  less  than  — . 

l^rst,  (12)2  =  144 ;     and     144  x  59  =  8496. 

Now,  the  square  root  of  8496  to  the  nearest  unit,  is  92 :  hence 

92  1 

—  =  7^^,   which   is   true   to   within   less   tkan    -—. 

2.  Find   the  y/^  to   within  less  than  — .  Ans.  3^. 
8.  Find  the  ^223  to  within  less  than  — .         Ans.  14fJ. 


CHAP,   v.]  SQUARE  ROOT  OF  FRACTIONS.  127 

97*.  The  manner  of  determining  the  approximate  root  in  deci- 
mals, is  a  consequence  of  the  preceding   rule. 

To   obtain   the  square   root  of   an   entire   number   within   --r, 

-— ,  &c.,  it  is  only  necessary,  according  to  the   preceding 

rule,  to  multiply  the  proposed  number  by  (lO)^,  (lOO)^,  (lOOO)^ ; 
or,  which  is   the   same   thing. 

Annex  to  the  number^  two,  four,  six,  dc,  ciphers :  then  extract 
the  root  of  the  product  to  the  nearest  unit,  and  divide  this  root 
hy  10,  100,  1000,  &c.,  which  is  effected  hg  pointing  off  one,  two, 
three,  c&c,  decimal  places  from    the   right   hand, 

EXAMPLES. 

1.  To  find  the   square  root  of  7   to  within  less   than  -rrjr. 

Having  multiplied  by  (100)^,  that  is, 
naving  annexed  four  ciphers  to  the  right 
hand  of  7,  it  becomes  70000,  whose 
root  extracted  to  the  nearest  unit,  is  264, 
which  being  divided  by  100  gives  2.64 
for  the  answer,  and  this  is  true  to  within 


7  0000 

4 

46 


2.64 


300 
276 


524 


2400 
2096 


less  than    -j^.  ^  304  Rem. 

2.  Find  the  V29  to  within  less  than   —— .  Ans.  5.38. 

3.  Find  the  ^227  to  within  less  than  rrjrjr^.    Ans.  15.0665. 


10000 

Remark. — The  number  of  ciphers  to  be  annexed  to  the  whole 
number,  is  always  double  the  number  of  decimal  places  required 
to   be  found  in  the  root. 

98.  The  manner  of  extracting  the  square  root  of  a  number 
containing  an  entire  part  and  decimals,  is  deduced  immediately 
from   the   preceding   article. 

Let   us   take  for  example   the  number  3.425.     This  Is  equiva- 

3425 
lent  to  .     Now,  1000  is  not  a  perfect   square,  but   the  de- 


128  ELEMENTS  OF  ALGEBRA.  [CHAP.   V. 

nominator  may  be  made   such  without  altering  the  value  of  the 
fraction,  by  multiplying  both   terms   by  10  ;   this   gives 

34250  34250 

10000  (100)2 

Then,  extracting  the  square  root  of  34250  to  the  nearest  unit, 

we  find  185  ;   hence,     — —     or  1.85  is  the  required  root  to  with- 
in less  than     — r. 

If  greater  exactness  be  required,  it  will  be  necessary  to  annex 
to  the  number  3.425  as  many  ciphers  as  shall  make  the  num- 
ber of  periods  of  decimals  equal  to  the  number  of  decimal 
places  to  be  found  in  the  root.  Hence,  to  extract  the  square 
root  of  a  mixed  decimal  : 

Annex  ciphers  to  the  proposed  number  until  the  whole  number 
of  decimal  places  shall  be  equal  to  double  the  number  required  in 
the  root,  Then^  extract  the  root  to  the  nearest  unit^  and  point  off, 
from  the  right  hand,  the  required  number  of  decimal  places, 

EXAMPLES. 


1.  Find  the   y/ 3271.4707    to  within  less  than  .01. 

Ans.  57.19. 

2.  Find  the   y/ 31.027    to  within  less  than  .01.     Ans,  5.57. 

3.  Find  the    ^/b.OlOOl    to  within  less  than  .00001. 

Ans,  0.10004. 

99.  Finally,  if  it  be  required  to  find  the  square  root  of  a 
vulgar  fraction  in   terms   of  decimals : 

Change  the  vulgar  fraction  into  a  decimal  and  continue  the  di- 
vision until  the  number  of  decimal  places  is  double  the  number 
required  in  the  root.  Then,  extract  the  root  of  the  decimal  by  the 
last  rule, 

EXAMPi^m-  . 

11 

1.  Extract  the  square  root  of    f-    -to  within  less  than  .001 
^  14 

This  number,  reduced  to  decimals,  is  0.785714  to  within  less 
than  0.000001.     The   root   of  0.785714,  to  the  nearest  unit,  is 


CHAP,   v.]  SQUARE  ROOT   OF  ALGEBRAIC   QUANTITIES.  129 

886:  hence,  0.886  is  the  root  of    — -     to  within  less  than    001. 
2.  Find  the   y^2j|   to  within  less  than  0.0001.     ^7*5.1.6931. 

Extraction  of  the  Square  Boot  of  Algebraic  Quantities, 

100#  Let  us  first  consider  the  case   of  a  monomial. 

In  order  to  discover  the  process  for  extracting  the  square 
loot,  let  us   see   how  the  square   of  a   monomial   is  formed. 

By  the  rule  for  the  multiplication  of  monomials  (Art.  42), 
we  have 

{^a'^PcY  z=z  ^a^b^c  X  ba^^c  =  26a^b^c^  ; 
that    is,   in    order    to    square    a    monomial,   it    is   necessary   to 
square  its  co-efficient,  and  double  the  exponent  of  each  letter. 

Hence,  to  find  the  square  root  of  a  monomial, 

Extract  the  square  root  of  the  co-efficient  for  a  neio  co-efficient^ 
and  write  after  this,  each  letter,  with  an  exponent  equal  to  its 
original  exponent  divided  by   two. 


Thus,    .^/Ma^  =  SaW  ;     for,     Sa^'^  x  Sa^^  =  64:a%\ 
and,       ^626a^h^  =  25a¥c^  ;     for,     (25a6*c3)2  =  625a^^c^ 

From  the  preceding  rule,  it  follows,  that,  when  a  monomial 
is  a  perfect  square,  its  numerical  co-efficient  .is  a  perfect  square, 
and  every  exponent  an  even  number. 

Thus,  25a*i2  \^  q,  perfect  square,  but  98a6^  is  not  b,  perfect 
«iquare ;  for,  98  is  not  a  perfect  square,  and  a  is  affected  with 
«in   uneven   exponent. 

Of  Polynomials, 

101.  Let  us  next  consider  the   case   of  polynomials. 

Let  N  denote  any  polynomial  whatever,  arranged  with  refer- 
ence to  a  certain  letter.  Now  the  square  of  a  polynomial  is 
the  product  arising  from  multiplying  the  polynomial  hy  itself 
ence:  hence,  the  first  term  of  the  product,  arranged  with  refer- 
ence to  a  particular  letter,  is  the  square  of  the  first  term  of 
the  polynomial,    arranged    witn    reference    to    the    same  letter. 


130  ELEMENTS  OF  ALGEBRA.  [CHAP.   V, 

Therefore,  the   square   root  of  the   first   term  of  such  a  product 
will   be  the   first  term  of  the  required   root. 
.    Denote  this  term  by   r,  and  the  following  terms  of  the  root, 
arranged  with  reference  to  the  leading  letter  of  the  polynomial, 
by  r',   t'\  r"\   &;c.,   and  we   shall  have 

iV^  =  (r  +  r'  +  r"  +  r'"  +  &c.  ;)2 
or,  if  we  designate  the  sum  of  all  the  terms  of  the  root,  after 
the  first,   by   5, 

iV^  =  (r  +  5)2  =  ^2  +  2r5  +  52 

=  r2  +  2r  {r'  +  r7  +  r'"  +  &c.)  +  s^ 

If  now  we  subtract  r^  from  iV",  and  designate  the  remaindei 
by  R^  we  shall  have, 

iVr-  7-2  =  i2  =  2r  (r'  +  /'  +  r'"  +  &c.)  +  s^ 
which  remainder  will  evidently  be  arranged  with  reference  to 
the  leading  letter  of  the  given  polynomial.  If  the  indicated 
operations  be  performed,  the  first  term  2rr'  will  contain  a 
higher  power  of  the  leading  letter  than  either  of  the  following 
terms,  and  cannot  be  reduced   with   any  of  them.     Hence, 

If  the  first  term  of  the  first  remainder  he  divided  hy  twice  the 
first  term  of  the  root,  the  quotient  will  be  the  second  term  of 
the   root. 

If  now,   we  place  r  +  r'  =n, 

and  designate  the   sum   of   the    remaining  terms   of    the  root, 
r'\   r"\   &c.,   by  s\   we   shall  have 

iV  =  (71  +  s'Y  z=zn^  +  2ns'  +  s'\ 

If  now  we  subtract  n'^  from  iV,  and  denote  the  remainder 
by   E\   we   shall  have, 

N-n'^^R  =  2ns'  +  s"^  =  2{r  +  r')  (r"  +  r'"  +  &c.)  +  s'»; 
in  which,   if    we    perform   the    multiplications   indicated    in   the 
second  member,  the  term  2rr"  will  contain  a  higher  power  of 
the  leading  letter  than   either  of  the  following  terms,  and  can- 
not,  consequently,   be  reduced  with  any  of  them.     Hence, 

If  the  first  term  of  the  second  remainder  he  divided  hy  twia 
the  first  term  of  the  root,  the  quotient  will  he  the  third  term 
of  the  root. 


CHAP,   v.]    SQUARE  ROOT    OF  ALGEBRAIC  QUANTITIES.  131 

If  we  make 

r  +  r'  +  r''  =  n\     and     r'"  +  r^  +  &c.  =  s'\ 
we  shall  have 

N=  (n'  +s"Y  =  w'2  +  2n's''  +  s"^;   and 

]Sr-n'^  =  R"  =  2  (r  +  ^'  +  r")  {r"'  +  r^  +  &c.)  +  s"^. 

in  which,   if   we    perforin    the    operations    indicated,   the    term 

2rr'"   will   contain   a   higher  power   of  the    leading   letter  than 

any  following  term.     Hence, 

If  we  divide  the  first  term  of  the  third  remainder  by  twice 
the  first  term  of  the  root,  the  quotient  will  be  the  fourth  term 
of  the  root. 

If  we   continue   the  operation,  we   shall   see,   generally,   that 

The  first  term  of  any  remainder ^  divided  by  twice  the  first 
term   of  the  rooty   will  give   a   new   term   of  the   required   root. 

It  should  be  observed,  that  instead  of  subtracting  n"^  from 
the  given  polynomial,  in  order  to  find  the  second  remainder, 
that  that  remainder  could  be  found  by  subtracting  (2r  +  r')r' 
from  the  first  remainder.  So,  the  third  remainder  may  be  found 
by  subtracting  (2n  -f-  r")r"  from  the  second,  and  similarly  for 
the  remainders   which  follow. 

Hence,  for  the  extraction  of  the  square  root  of  a  polynomial, 
we  have   the  following 

RULE. 

I.  Arrange  the  polynomial  with  reference  to  one  of  its  letters, 
and  then  extract  the  square  root  of  the  first  term,  which  will  give 
the  first  term  of  the  root.  Subtract  the  square  of  this  term  from 
the  given  polynomial. 

n.  Divide  the  first  term  of  the  remainder  by  twice  the  first  term 
of  the  root,  and  the   quotient   will   be   the   second  term  of  the  root. 

III.  From  the  first  remainder  subtract  the  product  of  twice  the 
first  term  of  the  root  plus  the   second  term,  by  the  second  term. 

IV.  Divide  the  first  term  of  the  second  remainder  by  twice  the 
first  term  of  the  root,  and  the  quotient  will  be  the  third  term  of 
the  root. 


132  ELEMENTS  OF  ALGEBRA.  [CHAP.   V, 

V.  From  the  secmd  remainder  subtract  the  product  of  twice  the 
sum  of  the  first  and  second  terms  of  the  root,  plus  the  third 
term,  by  the  third  term,  and  the  result  will  be  the  third  remait^- 
de^  from  which  the  fourth  term  of  the  root  may  be  found  as 
before, 

VI.  Continue  the  operation  till  a  remainder  is  found  equal  to 
0,  or  till  the  first  term  of  some  remainder  is  not  divisible  by 
iioice  the  first  term  of  the  root.  In  the  former  case  the  root  found 
is  exact,  and  the  polynomial  is  a  perfect  square;  in  the  latter 
case,  it  is  an  imperfect  square. 

EXAMPLES. 

1.  Extract  the  square  root  of  the  polynomial 

49a262  _  24a63  +  25a4  --  SOa^  +  I6b\ 

First  arrange  it  with  reference  to  the  letter  a. 


25a*  -  SOa^  +  49a262  -  24a63  +  166* 
25a4 


R  =  -  30^3^  +  49a262  -  24a63  +  166^ 
-30a36+    9a262 


R'  =  +  40a262  -  24a63  +  166* 

-I-  40a262  -  24a63  +  166* 


R''=: 


5a2  -  3a6  +  462 


10a2 

-3a6 
-3a6 

-  30^36 +  9a262 

10^2 

-  Qab  +  462 
462 

40a262  -  24a63  +  166*. 


2.  Find  the  square  root  of 

a*  +  Aa^x  +  Qa'^x^  +  4ax^  +  x\ 

3.  Find  the  square  root  of 

a*  —  2a%  +  Sa'^x^  —  2aa;3  +  x\ 

4.  Find  the  square  root  of 

4x^  +  12:^5  +  5^4  _  2aj3  +  7a;2  «.  2a;  +  1. 

5.  Find  the  square  root  of 

9a*  -  12a36  +  28a262  -  16a63  +  166*. 

6.  Find  the  square  root  of 

*^5a*62  -  40a362c  +  76a262c2  -  48a62c3  +  3662c*  -  30a*6c  f  24M^bi^ 
—  36a26c3  +  9a*c2. 


CHAP,   v.]         EADICALS  OF  THE   SECOND  PEGREE.  133 

Eemarks  on  the  JEJxtr action  of  the  Square  Root  of  Polynomials, 

1st.  A  binomial  can  never  be  a  perfect  square.  For,  its  root 
cannot  be  a  monomial,  since  the  square  of  a  monomial  will 
be  a  monomial  ;  nor  can  its  root  be  a  polynomial,  since  the 
square  of  the  simplest  polynomial,  viz.,  a  binomial,  will  con 
tain   at  least  three   terms.     Thus,  an  expression  of  the  form 

a2±62 

can  never  be   a  perfect  square. 

2d.  A  trinomial,  however,  may  be  a  perfect  square.  If  so, 
when  arranged,  its  two  extreme  terms  must  be  squares,  and  the 
middle  term  double  the  product  of  the  square  roots  of  the  other 
two.  Therefore,  to  obtain  the  square  root  of  a  trinomial,  when 
it  is  a  perfect  square. 

Extract  the  square  roots  of  the  two  extreme  terms,  and  give  these 
roots  the  same  or  contrary/  signs,  according  as  the  middle  term  is 
vositive  or  negative.  To  verify  it,  see  if  the  double  product  of  the 
two  roots   is   equal  to    the   middle    term   of  the    trinomial. 

Thus,     9a^  —  48a*52  _^  Q^a^"^    is  a  perfect  square, 
for,        y/'O^  =  3a3 ;      and,    ^^'Sia^  =  -  8ab^  ; 
also,  2  X  3a3  x(--  8a52)  =  —  48a^52,     the  middle  term. 

But  4a2  +  Uab  +  962 

is  not  a  perfect  square :  for,  although  4^2   and    +  962  q^^q  pep. 

feet   squares,  having  for   roots   2a    and    36,   yet  2  X  2a  X  36   is 

not  equal  to   14a6. 

Of  Radical  Quantities  of  the  Second  Degree. 

102#  A  radical  quantity  is  the  indicated  root  of  an  imperfeefc 
power  of  the  degree  indicated.  Radical  quantities  are  some- 
times called  irrational  quantities,  sometimes  surds,  but  more 
commonly,  simply  radicals. 

The  indicated  root  of  a  perfect  power  of  the  degree  indi 
cated,  is  a  rational  quantity  expressed   under   a   radical  form. 


134  ELEMENTS  OF  ALGEBRA.*  [CHAP.   V. 

An  indii-;ated  square  root  of  an  imperfect  square,  is  called 
a   radical  of  the   second   degree. 

An  indicated  cube  root  of  an  imperfect  cube,  is  called  a  radi- 
cal of  the  third  degree. 

Generally,  an  indicated  n^^  root  of  an  imperfect  n^'^  power, 
is   called   a   radical   of  the   n*^  degree. 

Thus,    y^  \/^    ^^^   V^'     ^^^  radicals  of  the  second  degree ; 
IJ  4,  ^/Ts"  and  ^/TT,  are  radicals  of  the  third  degree; 

and  1/4^  V^    ^^^    \/^>  ^^^  radicals  of  the  n^'^  degree. 

.  The  degree   of    a    radical    is    denoted    by   the    index   of   the 
root. 

The  index  of  the  root  is  also  called  the  index  of  the  radical, 

103.  Since  like  signs  in  both  factors  give  a  plus  sign  in  the 
product,  the  square  of  —  a,  as  well  as  that  of  + «,  will  be 
a^ :  hence,  the  square  root  of  a^  is  either  +  «  or  —a,  x\lso, 
the  square  root  of  2^a%^  is  either  +  baW-  or  —  ^ab'^.  Whence 
we  may  conclude,  that  if  a  monomial  is  positive,  its  square  root 
may  be   affected   either  with  the   sign    +    or    —  ; 

thus,  y'^yo*  =  ±  3a^ 

for,    +  3a2   or    —  Sa^,  squared,  gives   9a^.     The  double   sign  ± 
with  which   the   root  is   affected,  is   read  plus  or  minus. 

If  the  proposed  monomial  were  negative^  it  would  have  nfc 
square  root,  since  it  has  just  been  shown  that  the  square  of  every 
quantity,  whether  positive   or  negative,  is  essentially  positive. 

Therefore,  such   expressions   as, 

y^=^,      ^-  4.a\      y-  8a2^, 
are   algebraic  symbols  which  indicate   operations  that   cannot   be 
performed.      They   are    called    imagina^^^y    quantities^    or    rather, 
imaginary  expressions^  and   are   frequently   met   with   in   the   so- 
lution  of  equations   of  the  second   degree.     Generally, 

Every  indicated  even  root  of  a  negative  quantity  is  an  imaginary 
expression. 

An  odd  root  of  a  negative  quantity  may  often  be  extracted 

Fo-  example,     y"^=^  =  -  3,     since     (-  3)3  =  -  27. 


CHAP,   v.]         KAPIOALS    ;P  THE  SECOND  DEGREE.  1S6 

Radicals  are  similar  when  they  are  of  the  same  degree  and 
the  quantity  under  the  radical   sign  is   the   same  in  both. 

Thus,  a^T  and  c^^/T,  are  similar  radicals  of  the  seoocx: 
degree. 

Of  the  Simplification  of  Radicals  of  the  Second  Degree. 

104t  Radicals  of  the  second  degree  may  often  be  simplified, 
and  otherwise  transformed,  by  the  aid  of  the  following  prin- 
ciples. 

1st.  Let  the   .Vo^    and   ,J~b,    denote  any  two  radicals  of  th<j 
second  degree,  and  denote  their  product  by  p\   whence, 
^Xy/b^zp     ....     (1). 

Squaring  both  members  of  equation  (1),  (axiom  5),  we  have, 

{^Yx{/bY=:p\ 

or, ab=p'^    -     -     '     -     (2). 

Extracting  the  square  root  of  both  members  of  equation  (2), 
(axiom  6),  we  have, 

yfab=p\ 
but  things  which  are  equal  to  the  same  thing  are  equal  to  each 
other,  whence, 

.yf^  y.  .Jh  —  J~c^ \   hence, 
The  product  of  the  square  roots  of  two    quantities  is   equal  to 
the  square  root   of  the  product  of  those  quantities. 

2d.  Denote  the  quotient  of  .Va    by  ,V^  by   q ;   whence, 

^=,  ■  ■  ■  ■  (1). 

Squaring  both  members  of  equation  (1),  we  find, 

or,  ■  ^  =  g^    .    .    .     .     (2). 

Extracting  the  square  root  of  both  members  of  ec^uation  (2), 
we  have. 


Vt==- 


186  ELEMENTS  OF  ALGEBRA.  [CHAP.   V. 

Things  which  are  equal  to   the   same  thing    aie  equal   to  each 
oilier,  whence, 

The   quotient  of  the  square  roots  of  two  quantities  is  equal  to 
ilte  square  root  of  the  quotient  of  the  same  quantities. 

105.  The  square  root  of  98a6*  may  be  placed  under  the  form 
y^98^  =  y/49Fx~2^ 
which,  from   the  1st  principle  above,  may  be  written, 

In   like   manner, 

^4:ba%^cH  =^9aWc'^  X  bhd  =z  2>ahc,JUd. 
yseio^Jv^  r=:yi44a26Vo  X  ^hc  =:  12a6Vy^6^. 
The   quantity  which    stands  without   the   radical   sign  is  called 
Uie  co-efficient  of  the   radical. 

Thus,    76^,   3a6c,    and    V^ab^^^    are   co-efficients  of  the  radicals. 
In  general,  to   simplify  a  radical  of  the   second   degree : 
I.  Resolve  the  quantity  under  the  radical  sign  into  two  factors^  one 
of  which  shall  be  the  greatest  perfect  square  which  enters  it  as  a  factor. 

II.    Write  the  square  root  of  the  perfect  square  before  the  radical 
sign,  under  which  place  the  other  factor. 

EXAMPLES. 

1.  Reduce  Jlba^bc     to   its   simplest  form. 

2.  Reduce  J  12Sb^a^d'^     to  its   simplest  form. 

3.  Reduce  ^  S2a%^c     to   its   simplest  form. 

4.  Redu.ce  VlSGo^J*?    to   its   simplest  form, 

5.  Reduce  y/T024a96V     to  its  simplest  form. 

6.  Reduce  J'12'^aWcH    to   its  simplest  form. 

If   the  quantity   under  the  radical   sign  is  a  polynomiaj,  we 
may   often   simplify  the   expression  by   the   same  rule. 


CHAP,   v.]         KADICALS   OF  THE   SECOND  DEGREE.  137 

Take,  for   example,  the  expression 


ya36  +  4a262  +  4a63. 
The   quantity  under  the  radical  sign  is  not  a  perfect  square : 
but  it  can  be  put  under   the  form 

ab  (a2  +  4ab  +  462), 
Now,  the  factor  within  the  parenthesis  is  evidently  the  square 
of    a  +  26,    whence  we   have 

y/a36  +  4a262  +  4a63  =  (a  +  2b)  ^^ 
105*»  Conversely,  we  may  introduce  a  factor  under  the  radical 
sign. 

Thus,  av^-/^^/&; 

which    by    article    104,    is    equal    to    ^  a^b.     Hence, 

The  co-efficient  of  a  radical  may  be  passed  under  the  radical  sign, 
as  a  factor^  by  squaring  it. 

The  principal  use  of  this  transformation,  is  to  find  an  ap- 
proximate value  of  any  radical,  which  shall  differ  from  its  true 
value,  by  less  than  1. 

For  example,  take  the  expression    6^13. 

Now,  as  13  is  not  a  perfect  square,  we  can  only  find  an  ap- 
pi'oximate  value  for  its  square  root ;  and  when  this  approximate 
value  is  multiplied  by  6,  the  product  will  differ  materially  from 
the   true  value  of  6^13.        But  if  we  write, 

6^13  z:.y/62x  13=y36x  13  =/468, 
we  find   that   the   square   root  of  468   is   the  whole  number  21, 
to  within   less   than  1.     Hence, 

6.^/T[3  =  21,     to  within  less  than  1. 
In   a   similar   manner  we   may  find, 

I2V7"    —  31,     to  within  less  than  1. 

Addition  and  Siibtraction, 

106i  In   order   to   add   or   substract  similar  radicals ; 

Add  or  subtract  their  co-efficients^  and  :  the  ^um  or  differ- 
ence  annex   the   common   radical. 


138  ELEMENTS  OF  ALGEBflA*  [CHAP.    V 

Thus,  3ay6"  +  Sc^T  =  (3a  +  5c)y/T; 

and  3ay^  —  5cy/T  =  (3a  —  5c)  ^6^ 

In   like   manner, 

7^2^  +  3  y^r.:  (7  +  3)^2^  =  10/2^  ; 
and  7^2^  -  3  y^  =  (^  -  3)^^  =    4y^. 

Two  radicals,  which  do  not  appear  to  be  similar,  may  becomi 
so   by  simplification  (Art.  104). 
For   example, 

y  48a62  +  6^75^  =:  46y/3a  +  bh,J^z=.  95^3^; 

Alsa,  2  ^45       -  3y~5"  =  6  ^  -  3  ^5"  =  3  ^5. 

When  the  radicals  are  not  similar,  their  addition  or  subtrac- 
tion  can  only  be  indicated. 

Thus,  to   add    3,VT  to    5y^,     we  write, 

5/^+  3/5. 

Multiplication  of  Radical   Quantities  of  the   Second  Degree, 

107«  Let  a^^fb  and  cJ~d  denote  any  two  radicals  of  the  second 
degree;    their  product  will  be  denoted  thus, 

which,  since  the   order  of  the  factors   may   be   changed   without 
altering  the  value  of  the  product,  may  be  written, 

axe  Xy^Xy^ 
The  product  ol  the  last  factors   from   the   1st   principle  of  Art. 
104,  is  equal  jo  ^  hd\   we   have,  therefore, 

ay^  X  f -V^  ==  ac^Jhd, 
Hence,  t    multiply  one    radical  of  the    second    degree  by  au 
other,  we  have  the  following 

RULE. 

Multiply  the  co-efficients  together  for  a  new  co-efficient ;  after  this 
write  the  radical  sign,  and  under  it  the  product  of  the  quantities 
under  both  radical  signs. 


CHAP    v.]        BAICICALS  OF  THE  SECOND  DEGREE.  139 

EXAMPLES. 

2.  2ay^  X  ^a^Tc   =   ^a^.^fbh'^  =  ^a^hc. 


3.  %a^  a2  _|.  52  X  _  Sa^a^  +  6^  =  -  Ga^  (««  +  62). 

Division  of  Radical    Quantities  of  the  Second  Degree. 

108i  Let  a,^b  and  CyTd  represent  any  two  radicals  of  the 
second  degree,  and  let  it  be  required  to  find  the  quotient  of  the 
first  by  the  second.     This  quotient  may  be  indicated  thus, 

^^,  which  IS  equal  to  —  X  ^ ; 

but  from  the  2d  principle  of  Art.  104, 

^-/^-      hence         «/"*-»•  '^ 


d  '  cy^        c  V    d 

Hence,  to  divide  one  radical  of  the  second  degree  by  another, 
we  have  the  following 

RULE. 

Divide  the  co-efficient  of  the  dividend  by  the  co-efficient  of  the 
divisor  for  a  new  co-efficient;  after  this,  write  the  radical  sign^ 
"placing  under  it  the  quotient  obtained  by  dividing  the  quantity  under 
the  radical  sign  in  the  dividend  by  that  in   the  divisor. 

For  example,   haJb  -    2b ^fc^  — ?\/ — ; 
^  ^  2o  V    c 

And,  12acy^667~  4c ^26  =  3a \/-^  =  3a y^."" 

109.  The  following  transformation  is  of  frequent  application  ia 
finding  an  approxim,ate  value  for  a  radical  expression  of  a  par- 
ticular  form. 

Having  giren  an  expression  of  the  form, 

a  a 

or 


i>  +  V^  P-^' 


140  ELEMENTS   OF   ALGEBRA.  [CHAP.    V 

in  which  c»  and  p  are  any  numbers  whatever,  and  q  not  a  per 
feet  square,  it  is  the  object  of  the  transformation  to  render  the 
denominator  a  rational  quantity. 

This  object  is  attained  by  multiplying  both  terms  of  the  jfrac- 
tion  by  p—.^/Yy  when  the  denominator  is  J?+-/^,  and  by 
p -f  V^,  when  the  denominator  is  p—^g]  and  recollecting 
that  the  sum  of  two  quantities,  multiplied  by  their  difference,  is 
equal  to  the  difference  of  their  squares :    hence, 

a  __  (^{P  —  V^)  _^  ai^p  —  -y/ q)  _  ap  —  a  ^J~q ^ 

p-\-^/l[       {P+'y/9){p-y/9)  P^-9  P'^-9 

a  _  ctjp  +  V^)  _  cijp  +  -y/g)  _  ap  +  a\/^ 

P-^       {P-^){P+^)  P^-Q  P^-9 

in  which  the  denominators  are  rational. 

As  an  example  to  illustrate   the  utility  of  this  method  of  ap. 

proximation,  let  it  be  required  to  find  the  approximate  value  of 

7 
the  expression   —.     We  write 

^  —  V  ^ 
7         _  7(3  +  V^  _  21  +  7  -/5 


3 -^5  9-5  4 

But,  '7  v^—  -/49  X  5  =z  ^245  =  15  to  within  less 


than  1.     Therefore, 

7  21  +  15    to  within  less  than  1 


3-/5"' 


=  9     to    within 


less  than  -— ;     hence,  9  differs  from  the  true  value  by  less  than 

one  fourth. 

If  we  wish  a  more  exact  value  for  this  expression,  extract  the 
square  root  of  245  to  a  certain  number  of  decimal  places,  add  21 
to  this  root,  ana  divide  the  result  by  4. 

Take  the  expression,  — — — =, 

and  find  its  value  to  within  less  than  0.01. 


OHAF.   T.]     EXAMPLES   IN   THE   CALCULUS   OF    RADICALS.      141 
We  have, 

./Tr+/3~  11-3  ""  8 

Now,  7y^ =y/55  X  49  =y^2695  =  51.91,  within  less  than  0.01, 
and      7y/l5=ry/T5x  49=^735    ==27.11;    -  -    .     ; 

therefore, 

1  ^fh       __  51.91  -  27.11  24.80 
yiT  +  /3                 8  8 

Hence,  we  have  3.10  for  the  required  result.     This  is  true  to 

within  less  than    — — . 
oUO 

By  a  similar  process,  it  may  be  found,  ihat, 

3_i_2i/7" 

~^     ^ — -=2.123,   is  exact  to  within  less  than  0.001. 

5/12-6/5 

Remark. — The  value  of  expressions  similar  to  those  above, 
may  be  calculated  by  approximating  to  the  value  of  each  of  the 
radicals  which  enter  the  numerator  and  denominator.  But  as 
the  value  of  the  denominator  would  not  be  exact,  we  could 
not  determine  the  degree  of  approximation  which  would  be 
obtained,  whereas  by  the  method  just  indicated,  the  denomina- 
tor becomes  rational^  and  we  always  know  to  what  degree  of 
accuracy  the  approximatic«i  is  made. 

PROMISCUOUS   EXAMPLES. 


1.  Simplify   /125:  Ans.   5/5. 

/50~ 

2.  Reduce  \J  t^    to  its  simplest  form. 

We  observe  that  25  will  divide  the  numerator,  and  hence, 


■4 


'25  X  2       ^     72 


147  V  147* 

Since  the  perfect  square  49  will  divide  14*7, 

/2_ 


V   147  ~      V  49  X  3  ""  7  V  a 


142  ELEMENTS  OF  ALGEB]^.  [CHAP.  V. 

Divide  the  coeflScient  of  the  radical  by  8,  and  miitiply  the  num 
ber  under  the  radical  by  the  square  of  3  ;   then, 


5     /¥         5      /l8         5     /— 
7  V  3-  =  2TV  T  =  21^ 


3.  Reduce    ^  ^%a^x     to   its   most   simple  form. 

Ans.  "laJ^, 

4.  Reduce     J  {x^  —  a^x^)     to   its   most   simple  form. 

5.  Required  the  sum  of    ^^72*    and    Vl28. 

Ans.   14  y£ 

6.  Required  the  sum  of    ,V27     and    ^147. 

Ans.  lOya 

/2"  /27" 

7.  Required  the  sum  of    \/ -^    and    \  /  — r. 

8.  Required  the  sum  of   SlJa^h     and     3y^646^ 

9.  Required  the  sum  of   9^^243    and     10^363. 


/  3  /  5 

10.  Required  the  difference  of    \J  -r    ^"^^    \/o7* 

11.  Required  the  product  of     5V^  and    3V¥. 

^^5.  30,/Ta 


2      /T"  3      /T^ 

12.  Required  the  product  of     ~7^  \l ~^    ^^^      TV  To" 

13.  Divide     6y^    by     3^51 


10' 
^...  1/35. 


14.  What  is  the  sum  of    y/48a62     ^^d     5/ 75a, 


15.  What  is  the  sum  of    /TSaSp"    and    /SOo^P; 

^715.  (3a26  +  5a5)^  2a6. 


CHAPTER  VI. 

EQUATIONS    OF   THE    SECCND   CBGttXC. 

110»  Equations   of   the    second  degree    may  involve  but  on§ 
unknown  quantity,  or  they  may  involve  more  than  one. 
We  shall  first  consider  the  former  class. 

Ill,  An  equation  containing  but  one  unknown  quantity  is 
said  to  be  of  the  second  degree,  when  the  highest  power  of 
the  unknown   quantity  in  any  term,  is  the  second. 

Let  us  assume  the   equation, 

-j-x^  —  ex  +  d  =  cx^  -\ — -x  +  a, 
0  a 

Clearing  of  fractions, 

adx'^  —  bcdx  +  bd^  =  bcdx^  -f*  b^x  +  abd 

transposing,     adx'^  —  bcdx'^  —  bcdx  —  b^x  =  abd  —  bd^ 

factoring,         {ad  —  bcd)x^  -—  (bed  +  b^)x  =  abd  —  bd^ 

dividing  both  members   by  the  co-efficient  of  x^, 

bed  +  62         abd  -  bd^ 


^  -  ic  —  - 

ad  -—  bed  ad  —  bed ' 

If   we  now   replace  the    co-efficient    of    x    by    2p,   and    the 
second   member  by   q,   we   shall   have 

a;2  +  2px  =  q  ; 
and  since  every  equation  of  the  second  degree  may  be  reduced, 
in  like  manner,  we  conclude  that,  every  equation  of  the  second 
degree,  involving  but  one  unknown  quantity,  can  be  reduced  to 
the  form 

x^  +  2px  =  q, 
by  the  following 


144  ELEMENTS   OF  ALGEBR^  [CHAP.  VI. 

RULE. 

I.  Clear   the  equation   of  fractions ; 

II.  Transpose  all  the  known  terms  to  the  second  member^  and 
all   the   unknown   terms    to    the  first, 

III.  Reduce  the  terms  involving  the  square  of  the  unknown 
quantity  to  a  single  term  of  two  factors^  one  of  which  is  the 
square  of  the  unknown   quantity ; 

IV.  Then,  divide  both  members  by  the  co-efficient  of  the  square 
of  the   unknown   quantity, 

112.  If  2p,  the  algebraic   sum  of  the   co-efficients  of  the  first 

powers  of  x,  becomes   equal   to   Oj    the   equation  will   take  the 

form 

x^  =  q, 

and  this  is    called,  an   incomplete  equation  of  the   second  degree. 
Hence, 

An  incomplete  equation  of  the  second  degree  involves  only  the 
second  power  of  the  unknown  quantity  and  known  terms,  and  ma^ 
be  reduced  to  the  form 

x"^  =z  q. 

Solution  of  Incomplete  Equations,    ' 

113.  Having  reduced  the  equation  to  the  required  form,  we 
have  simply  to  extract  the  square  root  of  both  members  to  find  the 
value  of  the  unknown  quantity. 

Extracting  the  square   root  of  both   members  of  the   equation 

aj2  =  2',     we  have    x  =  V^ 

If  3'  is  a  perfect  square,  the  exact  value  of  x  can  be  found 
by  extracting  the  square  root  of  q,  and  the  value  of  x  will  then 
be  expressed  either  algebraically  or  in  numbers. 

If  q  is  an  algebraic  quantity,  and  not  a  perfect  square,  it  must 
be  reduced  to  its  simplest  form  by  the  rules  for  reducing  radi- 
cals of  the  second  degree.  If  g  is  a  number,  and  not  a  perfect 
square,  its  square  root  must  be  determined,  approximately,  by 
the  rules  already  given. 


CHAP.   VI.J       EQUATIONS   OF  THE   SECOND   DEGREE.  145 

But  the  sqitare  of  any  number  is  -|-,  whether  the  number 
itself  have  the    +    or    —    sign ;   hence,  it  follows  that 

(+/?)' =  !7.      and      (,-^Y^q; 
and  therefore,  the  unknown  quantity  x  is  susceptible  of  two  dis- 
tinct values,  viz : 

«=+V^   and    a;=-y^; 
and  either  of  these  values,  being  substituted   for   a;,   will  satisfy 
the  given  equation.     For, 

and  x^  =  — -/^  X  —  V^=  q',        hence, 

Every  incomplete  equation  of  the  second  degree  has  two  roots 
which  are  numerically  equal  to  each  other;  one  having  the  sigth 
plus,  and  the  other  the  sign  minus  (Art.  77). 


EXAMPLES. 


1.  Let  us  take  the  equation 

3  ^12  24  ^24 

which,  by  making  the  terms  entire,  becomes 

8^2  _  72  +  10a;2  =  7  -  24a;2  +  299, 
and  by  transposing  and  reducing 

42a;2  =  378      and    x^  =  ~  =  9 ; 
42 

hence,  x  =  +  V9"=  +  3;      and      x  =  ■— ,V^=  —  3. 

2.  As  a  second  example,  let  us  take  the  equation 

Sx^  =  5. 
Dividing  both  members  by  3  and  extracting  the  square  root, 

fe  which  the  values  of  x  must  be  determined  approxin:a</eIy 

3.  What  are  the  values  of  x  in  the  equation 

n{x^  -  4)  =  5(a^2  +  2).  Ans,  xz=:  ±^. 

4.  What  are  the  values  of  x  in  the  equation 

-i/m^  —  x^  .  m 
=  n,                  Ans  X  =  dz  — , 


146  ELEMENTS   OF  ALGEBRA.*  [CHAP.   VI. 

Solution    of  Equations  of  the  Second  Degree. 

114«  Let  us  now  solve  the  equation  of  the  second  degree 

x^  -\-  2px  =q.  i 

If  we  compare  the  first  member  with  the  square  of 
x.-^p^  which  is  x^ -{■2px -\- p^, 
we  see,  that  it  needs  but  the  square  of  p  to  render  it  a  perfect 
square.  If  then,  p^  be  added  to  the  first  member,  it  will  be 
come  a  perfect  square ;  but  in  order  to  preserve  the  equality  of 
the  members,  p^  must  also  be  added  to  the  second  member. 
Making  these  additions,  we  have 

x^  +  ^P^  -i-p^  =  q  +  p^  ', 
this  is  called,  completing  the  square^  and   is  done,  5y   adding   the 
square  of  half  the    co-efficient  of  x    to  both  members  of  the   equa 
tion. 

Now,  if  we  extract  the  square  root  of  both  members,  we  have, 

x-\-pz=z  ±y^gT^, 
and  by  transposing  p,  we  shall  have 

x  —  —p  -{-^q  +i>^     and     X  =  —p  —^q-\-p^. 
Either  of  these  values,  being  substituted  for  x  in  the  equation 

x^  +  2px  =  q 
will  satisfy  it.     For,  substituting   the  first  value, 

x'^  =  {—p  +^q+p'^Y  —f'  —  ^Py/~q~+¥  +  S'  -+  z?^ 

and 

2px  =z2px{-p  +yY+^)  =  -  2p2  +  2p^q+p% 
by  adding  x^  +  2px  =  q, 

iSubstituting  the   second  value  of  x,  we  find, 

a;2  =  (  —p  —^q-\.p^Y  —        p2  j^  2py^g'+^2^-  q -¥  T\  . 
and 

2px  ==2p{-'p  -y^TTF)  =  -  2p2  -  2p^q+p^  ; 
by  adding  x"^  +  2px  =  q  ; 

and   consequently,   both  values  found    above,   are    roots  of   the 
equation. 


CHAP.  VI.]       EQUATIOlsrS  OF  THE  SECOND  DEGREE.  147 

In  order  to  refer  readily,  to  either  of  these  values,  we  shall 
call  the  one  which  arises  from  asing  the  +  sign  before  the 
radical,  the  first  value  of  rr,  or  the  first  root  of  the  equation; 
and  the  other,  the  second  value  of  a?,  or  the  second  root  of  the 
equation. 

Having  reduced  a  complete  equation  of  the  second  degree  to 
the  form 

x^  +  2px  =  q^ 
we  can  write  immediately  the  two  values  of  the  unknown  quan 
tity  by  the  following 

RULE. 

I,  The  first  value  of  the  unknown  quantity  is  equal  to  half 
the  co-efficient  of  or,  taJcen  with  a  contrary  sign^  plus  the  square 
root  of  the  second  member  increased  by  the  square  of  half  this 
co-efficient. 

II.  The  second  value  is  equal  to  half  the  co-efficient  of  Xj 
tOjken  with  a  contrary  sign,  minus  the  square  root  of  the  s€C07id 
member  increased  by   the  sqvAire  of  half   this  co-efficient, 

EXAMPLES. 

1.  Let  us  take  as  an  example, 

a;2  -  7a:  +  10  =  0. 
Reducing  to  required  form, 

a;2  -  7a;  =  -  10  ; 


whence  by  the  rule,    a;  =  —  +  W  — -  10  H =  5  ; 

7  /  4Q 

and,     ^  a:  =  ~~^-10  +  ~  =  2. 

2,  As  a  second  example,  let  us  take  the   equation 


1  is  ELEMENTS  OF  ALGEBRA.*  [CHAP.  Vt 

Eeducing  tc    the  required  form,  we  have, 
„  .    2  360 


whence.  *=  "4 +\/^  +  ©' 


i 


It  often  occurs,  in  the  solution  of  equations,  that  p'^  and  q 
are  fractions,  as  in  the  above  example.  These  fractions  most 
generally  arise  from  dividing  hj  the  co-efficient  of  x^  in  the 
reduction  of  the  equation  to  the  required  form.  When  this  is 
the  case,  we  readily  discover  the  quantity  by  which  it  is  neces- 
sary to  multiply  the  term  q,  in  order  to  reduce  it  to  the 
same  denominator  with  p^ ;  after  which,  the  numerators  may  be 
added  together  and  placed  over  the  common  denominator. 
Afler  this  operation,  the  denominator  will  be  a  perfect  square, 
iind  may  be  brought  from  under  the  radical  sign,  and  will 
become   a   divisor   of  the   square   root   of  the   numerator. 

To  apply  these  principles  in  reducing  the  radical  part  of  the 
values  of  x,  in   the  last  example,  we  have 


7920  +  1 


7360      / 1  y_   .      7360x22        T^_  / 

V  22   ^  V22/  ""      V      (22)2     -1-  (22)2  y      (22)2 

and  therefore,  the  two  values  of  x  become, 

^"^       22  ^22  ""22""     ' 

1        89  90  45 

^^^  ^=-22-"2  =~22=~n' 

either  of  which  being  substituted  for  x  in  the  given  equation, 
will   satisfy  it. 

3.  What  ar«   the  values  of  a;  in  the  equation 
ax'^  —  ac  =  ex  —  Ix^ 


CHAP.   VI.]       EQUATIO:^rS  OF  THE  SECOND  DEGREE.  149 

Reducing  to  required  form,  we  have, 

c  ac 


a  +  b         a  +  b' 


whence,    ^  =  +  .^^J-^  +  ,/^ 


and,  ^=+ir77VlT-\/^^  + 


r2 


2  (a  +  6)       V  a  +  6    '   4  (a  +  6)2 
Eeducing    the    terms    under    the    radical    sign   to   a   common 

denominator,  we  find, 

/~ac  c2        __      /4a^c  4-  tahr  -\-c^  _  y'4a2c  -f  4abc  +  c^  . 

V^+^"'"4(a  +  6)2-V~i|^a  +  6)2       "  2(a  +  6) 

cdb  -t/  4a2c  +  4a5c  +  c^ 
hence,  «  = ^(^-j-^) ' 

4.  What  are  the  values  of  x,  in  the   equation, 

6a;2  _  37a;  =  -  57. 
By  reducing  to  the  required  form,  we   have, 

,      37  57 

x^--^x=-^-^, 


,   37         /     57   ,   /37\2 
wnence,  ,=  +  __±^^_  +  y 

Reducing  the  quantities   under  the  radical  sign  to  a  common 
denominator,  we  have, 


__       37  /-~114x  12       (37)2 

'^'■""^12      V         (12)2        +(12)2- 

But,    114  X  12  =  1368 ;   and   (37)2  =  1369 ; 


,            ,37^  /- 1368  +  1369    .  37  _^  1 
hence,     -=+j^^\/ ^y =  +  12=^12^' 

.  37  .1   19 
^=+l2+T2=-6-' 

.  S7   1   „ 

and.      ic  =  H =  3. 

'         ^12   12 

5.  What  are  the  values  of  x,   in   the   equation, 
4a2  -  2x'''  +  2ax  =  18a5  -  18^2. 


150  ELEMENTS  OF   ALGEBRA.  [CHAP.   VX 

Reducing   to   the   required  form,  we  have, 
x^  —  axzzz  2a^  —  9ab  +  9b^ ; 

whence,         ;t  =  -|-  db  1/2^2  —  9ab  +  9^2  +^ 

The  radical  part  is  equal  to    — —  36 ;     hence, 

«    .   /3a      ^,.  {x=z      2a  — 36. 

-^^^{-^-m;     or      \^^_  ^^g^_ 
Find  the  values  of  a;  in  the  following 

EXAMPLES. 

,      x^        a  ^        b  2x^  .  a  b 

i.     ~  — -7-a;=l X --.       Ans,    a;  =  ~,     a?  = . 

06  a  3  b  a 

^      dx   ^    Sx^    ^    ^        1  +  c       x^   ^    X 
c  4  c  4:        d 

1  (if 

Ans,    ir  =  -7,     a;  = 

c?'  c 


X^  ^^      I      ^^  __    Q  ^^  ^ 

■4"""3"'*"8"'^T~"3"* 


^W5.  ^  =  2- )   ^  =  "^  e" 


4.   -.- 


7. 


90         90            27 
a;        a;+l~a;  +  2' 

-4w5.    «  =s  4,      a;  =  —  -— 

2a;  -  10      „      a;  +  3 

8  -  a;        ~  ~  a;  -  2' 

9 

a;2      ,       6-1    . 

^  H X,     Ans,  X  =  a,   ar  = 

a                            '0 

a  -  6      ,    3a;2       a^ 
C     '+    2          e^  = 

6  +  a      ,   a;2       ^2 

c      ^+2        c2- 

6  +  a              6  —  a 
Ans,    X  = ,     X  = . 

CHAP.   VI.]      EQUATIONS  OF  THE   SECOND  DEGREE  151 

8.     moj*  -+  mn  =  2m.^/n  x  +  nx^ 
Ans,    X  = 


1^2 


-/^-y^'  y^+y^' 


,  ,       6a2    .     h'^x       ab  -  2^2       Sa^ 

9      a^a:2 r-  + = o ^. 

c^  c  c^  c 


2a  —b  3a  +  26 

Ans,    X  = ,     of  =  - 


be 


.^       4x^   ,   2x   ^    ^^       ,^      3a;2   ,    58a; 
10,     -rr-  +  -^^  +  10  =  19  -  -—  +  — r-. 
7  7  7  7 


Ans.    X  =  9,     X  =z  —I, 

11. 

X  +  a       ,       a  —  X                         .                           /b  +  2 

b  z= — ; — .                      Ans.    X  —  ±.a\/ -. 

X  —  a              a  +  X                                                  \    b  —  2 

12. 

2a;  +  2  =  24  —  5a;  —  2a;2.           Ans.    x  =  2,     x  =z  —  —. 

13. 

a;2  —  a;  —  40  =  170.             Ans.    x  =  15,  and  x  =  —  14. 

14. 

3a;2  ^  2a;  -  9  =  76.             Ans.    a;  =  5,    and  a;  =  ~  5f. 

15. 

a2  +  62  -  2bx  +  x^=  ^. 

Ans.    X  =  -r r  (bn  ±  Ja^m^  +  62^2  _  a^/^2\ 

Problems  giving  rise  to  Equations  of  the  Second  Degree  involv- 
ing hut  one  unknown  quantity, 

1.  Find  a  number  such  that  three  times  the  number  added  to 
twice  its  square  will   be  equal  to  65. 
*  Let  X  denote  the  number.     Then  from  the  conditions, 

2aj2  +  3a;=65     -     -    -     (1) 


Whence,  ^  =  ""-4^\/ Y  +  ^' 

reducing  a;  =n  5     and     «  =  — --. 


152  ELEMENTS   OF  ALGEBRAk  [CHJ^P    IV 

Both  of  these  roots  verify  the  equation:  for, 

2  X  (5)2  +  3  X  5  =  2  X  25  +  15  =  65; 

c  I     13\^  .   o  13       169       39       130       ^, 

and        2(--)+3x-^=e_--^^_  =  65. 

The  first  root  satisfies  the  conditions  of  the  problem  as  enutt 
eiated. 

The  second  root  will  also    satisfy  the  conditions,  if  we  regard 

its  algebraic  sign.      Had  we   denoted   the   unknown  quantity  by 

—  ic,   we  should  have  found 

2a;2-3a;  =  65     -     -     -     (2) 

13 
from  which  a;  =  —     and     a;  =  —  5. 

We  see   that   the    roots  of  this   equation   differ  from  those  of 

equation  (I)  only  in    their   signs,    a   result    which   was  to   have 

been   expected,  since   we   can   change   equation  (1)  into  equation 
(2)  by  simply  changing  the  sign  of  x^  and  the  reverse. 

2.  A  person  purchased  a  number  of  yards  of  cloth  for  240 
cents.  If  he  had  received  three  yards  less,  for  the  same  sum,  it 
would  have  cost  him  4  cents  more  per  yard.  How  many  yards 
did  he  purchase? 

Let  X  denote  the  number  of  yards  purchased. 

240 

Then   will  denote  the  number  of  cents  paid  per  yard. 

Had  he  received  three   yards   less, 
ic  — -  3,  would  have  denoted  the  number  of  yards  purchased,  and 

240 

5,  would  have  denoted  the  number  of  cents  he  paid  per  v  ai  d, 

X  —  o 

From  the   conditions  of  the   problem, 

240         240 


--:r  =  4; 


a?  —  3         X 

by  reducing,  a;^  —  3a;  =  180 

whence,  a?  =  15         and         a:  =  — 12. 

The   value     a;  =  15     satisfies   the   conditions   cf   the    pi  obi  em, 
understood  in  their  arithmetical   sense;   for,    U    yards  for  240 


CHAP.   VI.]       EQUATIONS  OF  THE  SECOND  DEGREE.  153 

240 

cents,  gives     •— — ,     or  16  cents  for  the  price  of  one  yard,  and 
Id 

12  yards  for  240   cents,   gives   20   cents   for   the  price   of  one 
yard,  which  exceeds   16  by   4. 

The  value  +a;=— 12,  or  --a;=:-:f-12,  will  satisfy  the 
conditions   of  the   following  problem : 

A  person  sold  a  number  of  yards  of  cloth  for  240  ^.ents : 
if  he  had  received  the  same  sum  for  "3  yards  more,  it  would 
have  brought  him  4  cents  less  per  yard.  How  many  yards  did 
he  sell? 

If  we  denote  the  number  of  yards  sold  by  a?,  the  statement  of  this 
last  problem,  and  the  given  one,  both  give  rise  to  the  same  equation, 

x^  —Sx  =  180, 

hence,  the   solution   of  this   equation   ought  to   give  the  answers 
to   both   problems,   as  we  see  that  it  does. 

Generally,  when  the  solution  of  the  equation  of  a  problem 
gives  two  roots,  if  the  problem  does  not  admit  of  two  solu- 
tions there  is  always  another  problem  whose  statement  gives 
rise  to  the  same  equation-  as  the  given  one,  and  in  this  case 
the  two   roots  form   answers  to   both  problems. 

3.  A  man  ^bought  a  horse,  which  he  sold  for  24  dollars.  At 
the  sale,  he  lost  as  much  per  cent,  on  the  price  of  his  pur- 
chase, as   the  horse  cost  him.     What  did  he  pay  for  the  horse? 

Let   X   denote  the  number  of  dollars  that  he  paid  for  the  horse  : 

then,    a;  —  24   will  denote  the  number  of  dollars  that  he  lost. 

But   as   he  lost  x  per   cent,  by   the   sale,  he   must  have  lost 

-—    upon  each  dollar,  and  upon  x  dollars   he    lost    a  numbei 

x^ 
of  dollars   denoted  by     jtw^;    we  have  then   the   equation 

:=x  —  24:,      whence      x^  —  100a?  =  —  2400  ; 

Therefore,  a:  =;  60       and      x  =*40. 

Both  of  these  values   satisfy  the  conditions  of  the  problem. 


154  ELEMENTS  OF  ALGEBgA.  LCHAP.   VL 

For,  in   the  first   place,  suppose   the  man  gave  60  dollars  for 
the   horse   and  sold   him  for  24,  he  then  loses  36  dollars.     But, 
from  the  enunciation,  he  should  lose  60  per  cent,  of  60,  that  is, 
60        .  _^         60  X  60      _ 

Too    "^^^   =-100-  =  ^^' 

therefore,  60  satisfies   the  problem. 

If  he  pays  40  dollars  for  the  horse,  he  loses  16  by  the  sale ; 
for,  he   should  lose  40  per  cent,  of  40,  or 

40X-^  =  16; 

therefore,  40  satisfies  the  conditions  of  the  problem. 

4.  A  grazier  bought  as  many  sheep  as  cost  him  £60,  and 
afler  reserving  15  out  of  the  number,  he  sold  the  remainder 
for  £54,  and  gained  25.  a  head  on  those  he  sold:  how  many 
did   he  buy?  Ans,  75. 

5.  A  merchant  bought  cloth  for  which  he  paid  £33  155.,  which 
he  sold  again  at  £2  85.  per  piece,  and  gained  by  the  bargain 
as  much  as  one  piece  cost  him :  how  many  pieces  did  he  buy  ? 

Ans.  15. 

6.  What  number  is  that,  which,  being  divided  by  the  product 
of  its  digits,  the  quotient  vdll  be  3 ;  and  if  18  be  added  to 
it,  the   order   of  its   digits   will   be   reversed?  Ans.  24. 

7.  Find  a  number  such  that  if  you  subtract  it  from  10,  and 
multiply  the  remainder  by  the  number  itself,  the  product  will 
be   21.  Ans.  7  or  3. 

8.  Two  persons,  A  and  B,  departed  from  different  places  at 
the  same  time,  and  traveled  towards  each  other.  On  meeting, 
it  appeared  that  A  had  traveled  18  miles  more  than  B ;  and 
that  A  could  have  performed  B's  journey  in  15f  days,  but  B 
would  have   been   28   days   in   performing    A's    journey.      How 

•I 


far  did  each   travel  ?  j  A  72  miles. 

B  54  miles. 


9.  A    company   at    a    tavern    had  £8  155.   to    pay   for    their 
reckoning ;    but  before  the    bill   was   settled,   two   of  them  left 


CHAP.   VI.]        EQUATIONS   OF  THE   SECOND   DEGREE.  155 

the  room,  and  then   those  who   remained  nad  lOs,  apiece  more 
to  pay   than  before :   how  many  were  there  in  the   company  ? 

Ans.  7. 

10.  What  two  numbers  are  those  whose  diiFerence  is  15,  and 
of  which  the  cube  of  the  lesser  is  equal  to  half  their  product  1 

Ans,  3  and  18. 

11.  Two  partners,  A  and  B,  gained  $140  in  trade:  A's  money 
was  3  months  in  trade,  and  his  gain  was  $60  less  than  his 
•'tock :  B's  money  was  $50  more  than  A'^  and  was  in  trade  5 
months  :  what  was  A's  stock  1  Ans,  $100. 

12.  Two  persons,  A  and  B,  start  from  two  different  points,  and 
travel  toward  each  other.  When  they  meet,  it  appears  that 
A  has  traveled  30  miles  more  than  B.  It  also  appears  that 
it  will  take  A  4  days  to  travel  the  road  that  B  had  come, 
and  B  9  days  to  travel  the  road  that  A  had  come.  What  was 
their  distance  apart  when  they   set  outi         Ans,   150  miles. 

Discussion  of  Equations  of  the    Second   Degree    involving  but 
one  unknown  quantity, 

115.  It  has  been  shown  that  every  complete  equation  of  the 
second  degree  can  be  reduced  to  the  form  (Art.  113) 

x^  +  2px=zq     .     -     -     (1), 

in   which  p   and   q   are   numerical   or   algebraic,   entire  or   frac- 
tional, and  their  signs  plus  or  minus. 

If  we  make  the  first  member  a  perfect  square,  by  completing 
the   square  (Art.  112*),  we  have 

x^  +  ^px  +  p^  =  q  +  p^, 
which  may  be  put  under   the  form 

{x+pY  =  q+pK 

Now,  whaj^aver  may  be  the  value  of  g  +  p^^  its  square  root 
may  be  represented  by  m,  and  the  conation  put  under  the  form 

(  X  +pY  =:  m^,     and  consequently ^     (s;  +  jp)^  —  m^  —  0. 


156  ELEMENTS  OF  ALGE'BI^.  [CHAP.   VI. 

But,  as  the  first  member  of  the  last  equation  is  the  differenca 
between   two   squares,  it  may   be  put  under  the  form 

{x-^-p  —m)  (rr  +  ^  +  ?7i)  =  0     .     -     -     (2), 
in  which  the  first  member  is  the  product  of  two  factors,  and  the 
second   0.      NotV,  we   can   make   this   product    eq^^al   to   0,   and 
consequently    satisfy    equation    (2)    only   in   two   different  ways . 
m.,  by  making 

^  + 1>  —  ^  =  0,     whence,     x  ■=.  —  p  +  m^ 
or,  by  making 

X  +  p  -\-  m  =.  0^     whence,     x  =  — p  — -  m. 
Now,  either  of  these   values  being   substituted  for   x  in   equa- 
tion (2),  will  satisfy  that  equation,  and  consequently,  will  satisfy 
equation  (1),  from  which  it  was   derived.     Hence,  we  conclude, 

1st.  That  every  equation  of  the  second  degree  has  two  roots,  and 
only  two. 

2d.  That  the  first  member  of  every  equation  of  the  second  degree^ 
whose  second  member  is  0,  can  be  resolved  into  two  binomial  fac- 
tors of  the  first  degree  with  respect  to  the  unknown  quantity,  having 
the  unknown  quantity  for  a  first  term  and  the  two  roots,  with  their 
signs  changed,  for  second  terms. 

For  example,  the  equation 

a;2  +  3^  _  28  =  0 
being  solved,  gives 

X  =:  4:     and     x  =  —  7 ; 

either  of  which  values  will  satisfy  the  equation.      We  also  have 
(a;  _  4)  (.^  4.  7)  3:3  a;2  +  3:r  -  28  =  0. 

If  the  roots  of  an  equation  are  known,  we  can  readily  form 
the  binomial  factors  and  deduce  the   equation. 

EXAMPLES. 

1.  What  are  the  factors,  and  what  is  the  equation,  of  which 
the   roots   are  8   and    —  9  ? 

Ans.         X  —  S    and    x  +  9        are  the  binomial  factors, 
and  x'^  +  x  —  12  =  0        is   ihe  equation. 


CHAr,    VI.]      EQUATIONS  OF  THE   SECOND  DEGREE.  157 

2.  What  are  the  factors,  and  what  is  the  equation,  of  which 
the  roots  are    —  1   and    +11 

a;  +  1     and    x  ■—  I  are  the  factors, 

and  a;2  —  1  =  0  is  the   equation. 

3.  What    are    the    factors,   and  what  is  the  equation,   whose 
roots   are 

7  +  -v/  -  1039         ,      7  _  y  -  1039  , 

/        7  +  V  -  1039\         ^      /        7  -  V  -  1039\ 
Ans.     ^x j     and      ^x ^j^ j 

are  the  factors, 

and  Sx^  —  7a;  +  34  =  0     is  the  equation. 

116»  If  we  designate  the  two  roots,   found  in  the  preceding 
article,   by  x'   and  x'\   we   shall   have, 

x^  =  —p  +  m, 
a;"  =  —  ^  —  m; 


or  substituting  for  m  its  value  ^  q  +  p^, 
x'  =  -p+^q+p2^ 

x"  =  -p-y/q+p^. 
Adding  these   equations,  member  to   member,  we  get 
x'  +  x"  =  —2^; 

and    multiplying    them,    member    by    member,     and    reducing, 

we  find 

ic'a;"  =  -^. 

Hence,  after  an  equation  has  been  reduced  to  the  form  of 

x^  +  2px  =  q^ 

1st.  The  ilgehraic  sum  of  its  two  roots  is^  equal  to  the  co-effir 
dent  of  the  first  power  of  the  unknown  quantity^  with  its  sign 
changed, 

2d.  The  product  of  the  *wo  roots  is  equal  to  the  second  member 
with  its  sign  changed. 


168  ELEMENTS  OF  ALGEBRA.  [CHAP.   VI. 

If  the  sum  of  two  quantities  is  given  or  known,  their  pro- 
duct  will   be   the   greatest  possible   when   they   are   equal. 

Let  2p  be  the  sum  of  two  quantities,  and  denote  their  differ- 
ence  by  2d'y   then, 
p  -\-  d    will  denote  the  greater,  and    p  ■—  d    the  less  quantity. 

If  we  represent  their  product  by    q^  we  shall  have 
p^  —  d'^  z=L  q. 

Now,  it  is  plain  that  q  will  increase  as  d  diminishes,  and 
that  it  will  be  the  greatest  possible,  when  c?  =  0 ;  that  is,  when 
the  two  quantities  are  equal  to  each  other,  in  which  ease  the 
product  becomes  equal  to  p'^.     Hence, 

3d.  The  greatest  possible  value  of  the  product  of  the  two  roots ^ 
is  equal  to  the  square  of  half  the  co-efficient  of  the  first  power 
of  the  unknown   quantity. 

Of  the  Four  Forms, 

II 7»  Thus  far,  we  have  regarded  p  and  q  as  algebraic  quan- 
tities, without  considering  the  essential  sign  of  either,  nor  have 
we   at   all   regarded   their   relative  values. 

If  we  first  suppose  p  and  q  to  be  both  essentially  positive, 
then  to  become  negative  in  succession,  and  after  that,  both  to 
become  negative  together,  we  shall  have  all  the  combinations 
of  signs  which  can  arise.  The  complete  equation  of  the  second 
degree  will,  therefore,  always  be  expressed  under  ora  of  the 
four  following  forms  : — 

x'  +  ^px^  q  (1), 
a;2  -2px=  q  (2), 
rc2  +  2p^  =  -  g  (3), 
a;2  —  2px  —  —  q  (4). 
These  equations  being   solved,  give 

^=-i'=ty~TTp      (t), 

^=+JP±/~~7+^         (2), 

«  =  -i^±y^=7Tp      (3), 

X=z  +p  :ty—  q  +  p^  (4). 


CHAP.  VI.]       EQUATIONS  OF  THE  SECOND  DEGREE.  159 

In  the  first  and  second  forms,  the  quantity  under  the  radical 
sign  will  be  positive,  whatever  be  the  relative  values  of  ^  and  g, 
since  q  and  p^  are  both  positive;  and  therefore,  both  roots 
will  be  real.     And  since 

g  -f  j92  yp2^     it  follows  that,      ^  q  ^^  P^  >  P, 
and  consequently,  the  roots  in  both  these  forms  will  have  the  same 
signs   as    the   radicals. 

In  the  first  form,  the  first  root  will  be  positive  and  the 
second  negative,  the  negative  root  being  numerically  the  greater* 

In  the  second  form,  the  first  root  is  positive  and  the  second 
negative,  the  positive  root  being  numerically  the  greater 

In  the  third  and  fourth  forms,  if 

the  roots  will  'r/o   real,  and  since 

they  will  have  the  same  sign  as  the  entire  part  of  the  root 
Hence,  both  roots  will  be  negative  in  the  third  form^  and  both 
'positive   in   the  fourth. 

If  ^2  __  q^  the  quantity  under  the  radical  sign  becomes  0, 
and  the  two  values  of  x  in  both  the  third  and  fourth  forms 
will  be  equal  to  each  other ;  both  equal  to  —  p  m  the  third 
form,  and  both   equal   to    +p  in   the   fourth. 

If  p'^  <  g,  the  quantity  under  the  radical  sign  is  negative, 
and  all  the  roots  in  the  third  and   fourth  forms  are  imaginary. 

But  from  the  third  principle  demonstrated  in  Art.  116,  the 
greatest  value  of  the  product  of  the  two  roots  is  p"^^  and  from 
the  second  principle  in  the  same  article,  this  product  is  equal 
to  q ;  hence,  the  supposition  of  p'^  <Cq  is  absurd,  and  the  values 
^  of  the  roots  corresponding  to  the  supposition  ought  to  be  im^ 
possible   or   imaginary. 

When  any  particular  supposition  gives  rise  to  imaginary  re- 
suits,  we  interpret  these  results  as  indicating  that  the  suppo 
sition  is  absurd  or  impossible. 


160  ELEMENTS  OF  ALGEBRA..  [CHAP.  VL 

If  p  =  0,  the  roots  In  each  form  become  equal  with  con- 
trary signs ;  real  in  the  first  and  second  forms,  and  imaginary 
in  the   third  and  fourth. 

If  q  =:  0,  the  first  and  third  forms  become  the  same,  as  also, 
the   second   and  fourth. 

In  the  former  case,  the  first  root  is  equal  to  0,  and  tfie 
second  root  is  equal  to  —  2p ;  in  the  latter  case,  the  first  root 
is   equal   to    +  2^,  and   the   second   to   0. 

If  ^  =  0  and  q  =  0,  all  the  roots  in  the  four  forms  reduce 
Xo   0. 

In  the  preceding   discussion  we  have  made 
_p2>g,  /^<g,       and      p^  =  q; 
we   have   also   made  p   and   q   separately   equal   to   0,  and  then 
both   equal   to   0   at  the   same   time. 

These  suppositions  embrace  every  possible  hypothesis  that  can 
be   made   upon  p  and  q, 

11 8t  The  results  deduced  in  article  117  might  have  been  ob- 
tiiined  by  a  discussion  of  the  four  forms  themselves,  instead  of 
their  roots,  making  use  of  the  principles  demonstrated  in  arti- 
cle 116. 

In  the  first  form   the  product  of  the   two   roots   is    equal   to 

—  g,   hence   the   roots   must  have   contrary    signs ;    their  sum  is 

—  2p,   hence  the  negative  root  is  numerically  the  greater. 

In  the  seco7id  form  the  product  of  the  roots  is  equal  to  —  q 
and  their  sum  equal  to  -\-  2p ;  hence,  their  signs  are  unlike, 
and  the   positive  root  is   the   greater. 

In  the  third  form  the  product  of  the  roots  is  equal  to  +  q ; 
hence,  their  signs  are  alike,  and  their  sum  being  equal  to  —  2p, 
they  are  both  negative. 

In  the  fourth  form  the  product  of  the  roots  is  equal  to  +  q, 
and  their  sum  is  equal  to  +  2p  ]  hence,  their  signs  are  alike 
and  both  positive. 

If  ^  =  0,  the  sum  of  the  roots  must  be  equal  to  0 ;  or  the 
roots   must  be  equal  with  contrary  signs. 


CHAP.   VI.]        EQUATIONS  OF  THE   SECOND  DEGREE.  161 

If  q  =  0,  the  product  of  the  roots  is  equal  to  0 ;  hence,  one 
of  the  roots  must  be  0,  and  the  other  will  be  equal  to  the  co- 
efficient of  the  first  power  of  the  unknown  quantity,  taken  with 
a  contrary  sign. 

If  ^  =:  0  and  q  =  0,  the  sum  of  the  roots  must  be  equal 
to  0,  and  their  product  must  be  equal  to  0 ;  hence,  the  root^- 
tliemselves  must  both  be  0. 

119.  Ihere  is  a  singular  case,  sometimes  met  with  in  the 
discussion  of  problems,  giving  rise  to  equations  of  the  second 
degree,  which  needs   explanation. 

To  discuss  it,  take  the   equation  x 

ax^  +  bx  =  c, 


which  gives  z  =  • 


2a 


If,  now,  we   suppose  a  =  0,   the  expression  for  the   value  of 

X  becomes 

0 

-^bdzb 


0 


whence, 


^  =  -0-' 


26 

X  =: =  00  . 

•  0 


But  the    supposition    a  =  0,   reduces    the   given   equation    to 
bx  —  c,  which  is  an  equation  of  the  Jirsi  degree. 

I'he   roots,  found   above,  however,  admit  of  interpretation. 

Tlie  first  one  reduces  to  the  form     —    in   consequence  of  the 

existence  of  a  factor,  in  both  numerator  and  denominator,  which 
factor  becomes  0  for  the  particular  supposition.  To  deduce  the 
true  value  of  the   root,   in   this   case,  take 

—  b  +  Wb'^  +  4ac 

T  1 

and  multiply  both  terms  of  the  fraction  by  —  b  —  J b'^ -\^ac\ 
aller  striking  out   the   common  factor  --  2a  we  shall   have 

_  2c 

~  b  +y^62-fW 
11 


162  ELEMENTS   OF  ALGEBgA.  [CHAP.   VI. 

ill    which,  if  we   make   a  =  0,    the   value   of    x  reduces   to     —; 

the    same  value   that   we    should    obtain   by   solving   the   simple 
equation   bx  =  c. 

The  other  root  od,  is  the  value  towards  which  the  expression, 
for  the  second  value  of  rr,  continaally  approaches  as  a  is  made 
smaller  and  smaller.  It  indicates  that  the  equation,  under  the 
supposition,  admits  of  but  one  root  in  finite  terms.  This  should 
be  the  case,  since  the  equation  then  becomes  of  the  first  degree. 

120*  The  discussion  of  the  following  problem  presents  most 
of  the  circumstances  usually  met  with  in  problems  giving  rise 
to  equations  of  the  second  degree.  In  the  solution  of  this 
problem,  we  employ  the  following  principle  of  optics,  viz. : — 

The  intensity  of  a  light  at  any  given  distance^  is  equal  to  its 
tjttensity  at  the  distance  1,  divided  by  the  square  of  that  distance. 

Problem  of  the  Lights, 


C"  A  C      B         C 

121#  Find  upon  the  line  which  joins  two  lights,  A  and  i?,  of 
different  intensities,  the  jfbint  which  is  equally  illuminated  by 
the   lights. 

Let  A  be  assumed  as  the  origin  of  distances,  and  regard  all 
distances  measured  from  A  to   the   right  as  positive. 

Let  c  represent  the  distance  AB^  between  the  two  lights  ; 
a  the  intensity  of  the  light  A  at  the  distance  1,  and  5,  the  in- 
tensity of  the   light  B  at   the   distance   1. 

Denote  the  distance  AO^  from  A  to  the  point  of  equal  illu- 
mination, by  x\  then  will  the  distance  from  B  to  the  same 
point  be   denoted  by   c  —  x. 

From  the  principle  assumed  in  the  last  article,  the  intensity 
of  the   light  -4,  at  the   distance   1,  being  a,  its  intensity  at   the 

distances   2,  3,  4,  &;c.,  will  be    — ,     — ,     — ,  &c. ;  hence,  at  thw 

d 

distance  x  it  will  be  expressed  \y^     ^. 


CHAP.   VI.]       EQUATIONS  OF  THE  SECOND  DEGREE.  163 

In   liKe   manner,  the   intensity  of  B  at   the  distance   c  — •  ar,  ia 
but,  by    the  conditions    of   the    problem,    these   two 


intensities  are  equal  to  each  other,  and  therefore  we  have  the 
equation 


a 


x^-  (c  -  xf  ' 
which  can  be  put  under  the  form 
(c  —  xY  _  b 
x^       ""   a  ' 

c  —  x       ±  ^/T         , 
tienoe,  = ^^  ;     whence 


c 


-y/'oT 


(1). 


(2). 


Since  both  of  these  values  of  x  are  always  real,  we  conclude 
that  there  will  be  two  points  of  equal  illumination  on  the  line 
A  B^  or  on  the  line  produced.  Indeed,  it  is  plain  that  there 
should  be,  not  only  a  point  of  equal  illumination  between  the 
lights,  but  also  one  on  the  prolongation  of  the  line  joining  tlie 
lights  and  on  the  side  of  the  lesser  one. 

To  discuss  these  two  values  of  x, 

First^   suppose    a  ^  b. 

The  first  value  of  x  is  positive;   and  since 

^        <1      . 

this  value  will  be  less  than  c,  and  consequently,  the  first  point  (7, 
will  be  situated  between  the  points  A  and.  B.  We  see,  moreover, 
that  the  point  will  be  nearer  B  than  A ;  for,  since  a^  o,  we 
have 

y^+y/a     or,     2ya>(y^+y^,     whence 

— ^ —  >  — ;     and  consequently,    —.r:^ 7=  >  — . 


164  ELEMENTS  OF  ALGEBKA.  [CHAP.   VL 

Tlie  second  value  of  x  is   also  positive;   but  since 

•v^     >1 

it  \*ill  be  greater  than  c;  and  consequently,  the  second  point 
wlL  be  at  some  point  C\  on  the  prolongation  of  AB^  and  at 
the  right  of  the,  two  lights. 

This  is  as  it ^ should  be;  for,  since  the  light  at  A  is  most 
intense,  the  point  of  equal  illumination,  between  the  lights,  ought 
to  be  nearest  the  light  B\  and  also,  the  point  on  the  prolonga- 
tion of  AB  ought  to  be  on  the  side  of  the   lesser  light  B, 

Second^    suppose     a  <^h. 
The  first  value  of  x   is  positive ;   and   since 

this  value  of  x  will  be  less  than  c;  consequently,  the  first  point 
will  fall  at  some  point  (7,  to  the  right  of  A^  and  between  A 
and  B, 


C"  A  C      B         C 

We  see,  moreover,  that  it  will  be  nearer  A  than  B\  for, 
since   a<^h^   we   have 

J~a+jTy  2  J  a,      and  consequently,  '^ — —  <  — . 

V^+V^        2 

The  second  value  of  x  is  essentially  negative,  since  the  nume- 
rator is  positive,  and  the  denominator  essentially  negative. 

We  have  agreed  to  consider  distances  from  A  to  the  right 
positive;  hence,  in  accordance  with  the  rule  already  established 
for  interpreting  negative  results',  the  second  point  of  equal  illu- 
mination will  be  found  at  C'\  somewhere  to  the  left  of  A. 

This  is  as  it  should  be,  since,  under  the  supposition,  the  light 
at  B  is  most  intense;  hence,  the  point  of  equal  illumination, 
between  the  two  lights,  should  be  nearest  A^  and  the  point  in 
the  prolongation  of  AB^  should  be  on  the  side  nearest  the 
feebler  light  A, 


CHAP.   VI.J       EQUATIONS  OF  THE  SECOND  DEGREE..  165 

Tliird^   suppose    a  =:b,    and    c  >  0. 
The  firs*,  value  of  x  is  then  positive,  and  equal   to   —   hence, 
the  first  point  is  midway  between  the  two  lights. 

The  second  value  of  x  becomes  =  oo ,  a  result  which  in- 

dicates  that  there  is   no    other   point  of  illumination   at   a   finite 
distance  from  A, 

This  interpretation  is  evidently  correct;  for,  under  the  suppo- 
sition made,  the  lights  are  equally  intense,  and  consequently,  the 
point  midway  between  them  ought  to  be  equally  illuminated. 
It  is  also  plain,  that  there  can  be  no  other  point  on  the  line 
which  will  enjoy  that  property. 

Fourth,   suppose     b  =^  a  and  c  =  0. 

The  first  value  of  x  becomes,  —  =  0,   hence   the  first  point 

is  at  A, 

The  second  value  of  x  becomes,    — ,    a   result    which   indicates 

that   there   are   an    infinite   number    of   other    points    which   arc 
equally  illuminated. 

These  conclusions  are  confirmed  by  a  consideration  of  the  con- 
ditions of  the  problem.  Under  this  supposition,  the  lights  are 
equal  in  intensity,  and  coincide  with  each  other  at  the  point  A. 
That  point  ought  then  to  be  equally  illuminated  by  the  lights, 
as  ought,  also,  every  other  point  of  the  line  on  whidi  the  lights 
are  placed. 

Fifth^    suppose     a  >  ^,    or   a  <^b^    and   c  =  0. 

Under  these  suppositions,  both  values  of  x  reduce  to  0,  which 
shows  that  both  points  of  equal,  illumination  coincide  with  the 
point  A, 

This  is  evidently  the  case,  for,  since  a  is  not  equal  to  5, 
and  the  lights  coincide  at  ^,  it  is  plain  that  no  other  point  than 
A  can  be  equally  illumina^d  by  them. 

The  preceding  discussion  presents  a  striking  example  of  the 
precision  with  which  the  algebraic  analysis  'respond  i  to  all  the 
'•elations  which  exist  between  the  quant'ties  that  enter  a  problem. 


166  ^  ELEMENTS  OF  ALGEBRA.  LCHAP.  VI. 


EXAMPLES   INVOLVING   RADICALS    OF   THE    SECOND    DEGREE. 

1.  Given,  x  -fVa^  -|-  x^  =  -,  to  find  the  values  of  «. 

^  ^a-^  +  x^ 

By  reducing  to  entire  terms,  we  have, 

x^a^-i-  x^  +  a^  +  x^  =  2a2', 
oj  transposing,  ajy^a^  -f  x^  =  a^  —  x^, 

and  by  squaring  both  members,      a^x'^  +  a;*  =  a*  —  2a2^2  _|_  ^^ 
whence,  Sa?x^  =  a*, 

and,  X  =  ±: 


2.  Given,  \/—^+b^-^  \  /  —-  —  P  =  b,  to  find  the  values  of  x» 
\  x^  \    x^ 

By  transposing,    y^ -^  +  b^  =</  —  —  b^  +  b -, 

squaring  both  members,  -r  +  ^^  =  —  —  i^  +  26  \  / 6^  4-  6^ ; 

x^        ■        x^  \   x^ 


whence,  b'^^^bJ—^-b\     and    5  =  2\/~-62; 

squaring  both  members,  b'^  =  -^  —  46^  ; 

and  hence,  a;^  =  — ,     and    a;  =  ± 


552  b^h 

d  /  Ci^  X^  X 

3.  Given, \-  \/ —  =  --.,  to  find  the  values  of  x, 

X         \       x^  b 

Ans,    x=  ±i  J  "Hab  —  6*. 

4.  Given, \/^  +  ?\/ — ^ —  =  ^H/ — 7 — »    to  ftnd  the 
values  of  ir.  .a 


CBAl\   VI.]        EQUATIONS  OF  THE  SECOND  DEGREE.  167 


a  —  -yd 


•^  —  x^ 


6.  Given,  ^  =  t,  to  find  the  ralues   of  x. 

a  •\-  J  d^  —  x^ 

2aVT 

Ans,  x=  ±:  - — --7-. 
1  -j-c> 

6.  Given,  ^^      ^ }=: ,  to  find  the  values  of  x. 

J  X  —Jx—ai       X  —a 
^        ^  ,  ail^ny 

7.  Given,  ■ h  •^^- — — —  z=z\J  —^  to  find  the  values  of  aj. 

^/i5.    a;  =  ±  2^a6  —  i^' 

8.  Given,  ^^r =  6,   to  find  the  values  of  x, 

'  a  +  X 


±  a(l  ± 'v/26"^^ 

Of  Trinomial  Equations, 

122«  JV^  trinomial  equation  is  one  which  .  involves  only  terms 
containing  two  different   powers  of  the   unknown  quantity  and  a 
;r-  known   term   or   terms. 

h^     123«  Every  trinomial   equation   can   be   reduced   to   the  form 

x"^  +  ^px""  —  q (1), 

in  which  m  and  n  are  positive  whole  numbers,  and  p  and  q 
known  quantities,  by  means  of  a  rule  entirely  similar  to  that 
given  in   article   111. 

If  we  suppose   m  =  2    and    ^  =  1,    equation   (1)   becomes 
x'^  +  ^px  =  5', 
a   trinomial   equation    of  the   second   degree. 

124.  The  solution  of  trinomial  equations  of  the  second  degree, 
has  already  been  explained.     The  methods,  there  explained,  are^ 
with  some  slight  modifications,  applicabfe  to  all  trinomial  eqm^ 
tions  in  which  m  =  2/i,  that  is,  to  all   equations  of  the  form 
x^^  +  2j0ir"  =  q. 


168  ELEMENTS   OF   ALGE^A.  [CHAP,    VI. 

To  demonstrate  a  rule  for  the  solution  of  equations  of  this 
form,  let  us  place 

ic**  =  y  ;       whence,       a:**  =  y^. 

Those  values  of  x"^  and  rr^",  being  substituted  in  the  given 
equation,  reduce   it   to 

whence,  y  =  ~  j?  db  Vg  4-/>2^ 

or,  a?"  z=  —  ^  dz  V  g  +  p*^. 

Now,  the  ^'^  root,  of  the  first  member,  is  x  (Art.  18),  and 
although  we  have  not  yet  explained  how  to  extract  the  ?i'* 
root  of  an  algebraic  quantity,  we  may  indicate  the  n^^  root  of 
the   second   member.     Hence,  (axiom  6), 

Hence,  to  solve  a  trinomial  equation  which  can  be  reduced 
to   the   form    x^^  +  2;pa;"  nr  q^    we   have   the  following 

RULE. 

Reduce  the  equation  to  the  form  of  or^"  -f-  2px^  ==  5'  /  the  values 
of  the  unknown  quantity/  ivill  then  he  found  by  extracting  the 
n*^  root  of  half  the  co- efficient  of  the  lowest  power  of  the  un- 
known quantity  with  its  sign  changed,  plus  or  minus  the  square 
root  of  the  second  member  increased  by  the  square  of  half  tlie 
co-efficient   of  the   lowest  poioer   of  the   unknown   quantity. 

If  n  =  2,  the  roots   of  the  equation  are  of  the  form 


^sj  -P  ±/7+P« 


We  see  that  the  unknov/n  quantity  has  four  valaes^  sirxe  each 
of  the  signs  +  and  — ,  which  affect  the  first  radical  can  be 
combined,  in  successicn,*with  each  of  the  signs  which  affect  the 
second ;  hut  these  values,  taken  two  and  two,  are  numerically  equals 
and  have  contrary  signs. 


CHAP.   VI.]  TRINOMIAL  EQUATIONS.  169 

EXAMPLES. 

1.  Take  the   equation 

Tliis  being  of  the  required  form,  we  have   by  appication  of 
tli^   rule, 


/25       7" 
whence,  a;=dzy— ±y; 

hence,  the  four  roots  are    +4,    —  4,    +3,    and    —  3. 

2.  As  a  second  example,  take  the  equation 

x^  -  7a;2  =  8. 
Whence,  by  the  rule, 

hence,  the  four  roots  are, 

+  2/2;  -2/2;  +/=n:  and  -/^=-r; 

the  last  two  are  imaginary. 

3.  a:4  _  ^2bc  +  4a2)  x^  =  -  bh\ 

Ans.  X  =  ii/^c  +  2a2  dz  2ay/^>c  +  a'^. 

4.  2a;  -  7y^  =  99.  ^715.   a;  =  81,  a;  =  1|1 

5.  4^5.r*  +  4.i,2^0.    .    Ans.x=±./l^V^^^II 

h  d  y  2bd 

125»  The  solution  of  trinomial  equations  of  the  fourth  deg.  '»i 
requires  the  extraction  of  the  square  root  of  expressions  of  tiiw 
form  of   a  dz  .^   in  which    a   and   h    %re   positive   or   negative, 

numerical  or  algebraic.  The  expression  \/  a  =h  ,^/T  can  some- 
times be  reduced  to  the  form  of  a'  db  .^Tb'  or  to  the  form 
^ a"  dz  ^J~b^\    and  when    such   transformation   is    possible,  it  Ls 


170  ELEMENTS   OF   ALGEBKA.  [CHAP.    VX 

advantageous   to    effect   it,  since,  in   this   case,  we   have  only  t<> 
extract  two  simple  square  roots ;  whereas,  the  expression 


y  «  ±V^ 


requires  the  extraction  of  the  square  root  of  the  square  root. 

To   deduce  forinulas  for   making   the    required  transformation, 
let  us  assume 


:p^rq- 


V^+V^    ....     (1), 


i^-^  =  \/«-V^  .     (2); 

in  which  ^  and  q  are  arbitrary  quantities. 

It   is   now  required   to  find   such  values  for  p  and   q   as  will 
satisfy  equations  (1)  and  (2). 

By  squaring  both  members  of  equations  (1)  and  (2),  we  have 
^2  +  2^^  +  ^2^a+yy.    .    .     (3), 
^2_2i?^  +  ^2^a~y6".     -    .     (4). 
Adding  equations  (3)  and  (4),  member  to   member,  we   get 

f'^q^=^a (5). 

Multiplying  (1)  and  (2),  member  by  member,  we   have. 

Let  us   now  represent  J  a^  —  h   by  c.     Substituting     in     the 
last  equation, 

f'--q^=.c (6). 

From  (5)  and  (6)  we  readily  deduce, 

these  values  sibstituted  for  p   and  q,  in  equations  (1)  and  (2), 
give 


vA^-^V^'^Vn 


CHAP.   VI. J  TRINOMIAL  EQUATIONS.  171 

hence, 

„d     /r7==t(v/5±i-^) . .  (9). 

Now,  if  o?  —  h  is  a  perfect  square,  its  square  root,  c,  will 
be  a  rational  quantity,  and  the  application  of  one  of  the  for- 
mulas (7)  or  (8)  will  reduce  the  given  expression  to  the  re- 
quired form.  If  o?  —  b  is  not  a  perfect  square,  the  applicatioi 
of  the  formulas  will  not  simplify  the  given  expression,  for,  we 
shall   JstiU   have   to   extract   the   square  root  of  a   square   root. 

Therefore,  in   general,  this   transformation  is   not  used,  unless 
a^  —b   is  a  perfect  square. 


EXAMPLES. 


1.  Reduce   v/^^  +  42^5  inW  94  +  ^8820,  to  its  simplesi 
form.     We  have,  a  =  94,     b  =  8820, 

whence,  c  =  ^  a^ —  b  =  ^  8836  — 8820  =  4, 

a  rational   quantity ;  formula   (7)   is   therefore  applicable  to  this 
case,   and  we  have 

or,   reducing,  =  ±  (y^49  +y^45)  ; 


hence,  ^94  +  42^  =  ±  (  7  +  3^5). 

This   may   be   verified;   for, 

(7  +  3y^)2  =  49  +  45  +  42y^=  94  +  42yA5. 

2.  Reduce      a/ nj^  +  2^^  —  ^mj^  +  m^,      to    its    simplest 

form.     Ye  haye 

(7  =  72J9  +  2771^,     and     b  =  Am^inip  +  ^\ 
a*  -  ^  =r:  w2p2^     and     c  =.Jo?-  —  6  =  wjt?; 


172  ELEMENTS   OF  ALGEBRA.  [CHAP.   VL 

and   therefore,   formula   (7)   is   applicable.     It  gives, 

=^lV 2 V 2 > 

and;   reducing,  ±  (*/  ^i?  +  ^^  —  ^). 

3.  Reduce   to   its   simplest  form, 


a/ 16  +  ZOyf^^  +  U  16  -  SO^/"^. 
By   applying   the   formulas,   we   find 


4/I6  +  SOyTIl  =  5  +  3yA31, 


and  W  16-S0ynri  =  S-Sy'^^^l: 

hencs,  a/ 16  +  80y^^  +  v/ ^^  -  ^^-/"^^  =  !<>• 

This   example   shows   that  the  transformation  is  applicable  to 
imaginary  expressions. 

4.  Reduce  to  its  simplest  form, 

1/28  +  lOy/3.  Arts.   5  +^3. 

5.  Reduce   to   its   simplest   form, 

\/l  -\-A,/^-^.  Arts.    2  +,^/rr3. 

6.  Reduce   to  its   simplest  form, 

Uhc  +  ^h^bc-b'^  -  ^/bc  -  2Sy  6c  -  62. 

^«5.     db2i 

7.  Reduce   to   its   simplest   form, 

4/06  +  4c2  ~  (^2  _  2^  ^abc^  -  abd'^, 

Ans,  J~ab  -     /4c^  —  o?* 


CHAP.   VI.]       EQUATIONS   OF   THE   SECOND   DEGREE.  173 

Equations  of  the  Second  Degree  involving  two  or  more  unknown 

quantities, 

126t  Every  equation  of  the  second  degree,  containing  two 
unknown  quantities,  is  of  the   general   form 

ay2  +  hxy  +  cx^  +  dij  -\-fx  +  ^  =  0 ; 
or   a  particular   case  of  that   form.     For,  this  equation  contains 
terms  involving   the   squares   of    both   unknown   quantities,    theii 
product,    their   first  powers,  and  a   known    term. 

In  order  to  discuss,  generally,  equations  of  the  second  degree 
involving  two  unknown  quantities,  let  us  take  the  two  equations 
3f  the   most  general   form 

ay"^  +  hxy  +  cx'^  +  dy  -\-fx-\-  g  =0, 
and  a'y'^  +  h'xy  +  c'x'^  +  d'y  -\-fx  +  </'  =  0. 

Arranging   them   with  reference   to   x,   they   become 
cx'^  +  {hy  +f)x  -\-ay'^  +  dy  +  g   =0, 
c'^2  +  (h'y  +f)  X  +  aY  +d'y  +  g'=zO', 
from  which   we  may  eliminate   ic^,  after  having  made  its  co-effi- 
cient  the   same   in   both   equations. 

By  multiplying  both  members  of  the  first  equation  by  c\  and 
both   members  of  the  second  by  c,   they  become, 

cc'x''  +{hy+f)c'x+{ay^  +  dy  +  gy=iO, 
cc'x^  +  [h'y  +f')c  X  +  {aY  +  d'y  +  g')c  =  0. 
Subtracting  one  from  the  other,  member  from  member,  we  have 
{{he'  —  cb')y  +fc'  —  cf'^x  +  (ao    —  ca^/  +  {dc'  —  cd')y  +  gc' 

~  eg'  =  0, 
which  gives 

__  {ca'  —  acyf  +  {cd'  —  dc')y  +  eg'  —  gc' 
^  '-  (b&  -  chyj  -^fc'  -  cf 

This  value  being  substituted  for  x  in  one  of  the  proposed 
equations,   will   give   a  final  equation^  involving   only   y. 

But  without  effecting  the  substitution,  which  would  lead  to  a 
very  complicated  result,  it  is  easy  to  perceive  that  the  final 
equation  involving   y,   will   be  of  the   fourth   degree.      For,   the 


174  ELEMENTS   OF   ALGEBR^.  [CHAP.'   VI 

numerator  of  the   value  of  x   being  of  the  form 

my^  +  ny  +  p^ 
its   square   will   be  of  the   fourth   degree,  and   this  square  forms 
one   of  the  parts   in   the  result  cf  the   substitution. 

Therefore,  in  general,  the  solution  of  t2vo  equations  of  the  secona 
degree^  involving  two  unknown  quantities^  depends  upon  that  of  an 
equation  of  the  fourth  degree^  involving  one  unknown  quantity, 

127.  Since  we  have  not  yet  explained  the  manner  of  solving 
equations  of  the  fourth  degree,  it  follows  that  we  cannot,  as 
yet,  solve  the  general  case  of  two  equations  of  the  second 
degree  involving  two  unknown  quantities.  There  are,  however, 
some  particular  cases  that  admit  of  solution,  by  the  application 
of  the   rules   already  demonstrated. 

First.  We  can  always  solve  two  equations  containing  two 
unknown  quantities,  when  one  of  the  equations  is  of  the  second 
degree,  and  the   other   of  the   first. 

For,  we  can  find  the  value  of  one  of  the  unknown  qua^i 
titles  in  terms  of  the  other  aiid  known  quantities,  from  the 
latter  equation,  and  by  substituting  this  in  the  former,  we  shall 
have  a  single  equation  of  the  second  degree  containing  but  one 
unknown   quantity,  which  can  be   solved. 

Thus,  if  we   have   the   two   equations 

a;2  +  2y2  ::zr  22  .     -     -      (1), 

2^   -    y   =    1     -    -     -    .     (2), 
we  can  find  from   equation  (2), 

-1+^-      whence,      .^  =  l±^±t , 


2 

and  by  substituting  this  expression  for  x^  in  equation  (1),  we  find 
-r  ^y  -ry_  _^  ^^^  ^  ^^  . 

whence  we  get  the  values  of  y :   that  is, 

29 
y  =  3      and      y  =  —  y  ; 

and  by  substituting  in  equation  (2)  we  find, 

r  =  2      and      a?  =  — ^. 


^flAP.   VI.]        EQUATIONS   OF  THE  SECOND  DEGREE.  175 

Second.  We  can  always  solve  two  equations  of  the  second 
degree  containing  two  unknown  quantities  when  they  are  boCh 
homogeneous  with  respect  to  these  quantities. 

For,  we  can  substitute  for  one  of  the  unknown  quantities, 
an  auxiliary  unknown  quantity  multiplied  into  the  second  un- 
known quantity,  and  by  combining  the  two  resulting  equations 
we  can  find  an  equation  of  the  second  degree,  from  which  the 
value  of  the  auxiliary  unkno^vn  quantity  may  be  determined, 
and  thence  the  values  of  the  required  quantities  can  easily  be 
found. 

Take,  for  example,  the  equations 

a;2+    ary—    2/2  —  5     ...     (i)^ 

3a;2  —  2a;y  —  2?/2  =  6     -     -     -     (2). 
Substitute  for  y,    px,  p   being   unknown,  the   given   equati-#ns 
become 

3a;2  —  2pa;2  -  2j92a;2  ™  6     -     -     -     (4). 
Finding   the  values  of  x^  in   terms  of   j9,    from  equations  (S) 
and   (4),  and  placing  them  equal  to   each   other,  we  deduce 
5  6 


\+p-p^        3-2p-2i32' 

or  reducing. 

^2  +  4p  =  _  ; 

whence, 

2?  =  — ,      and      i>  =  -  Y* 

Considering  the  positive  value  of  p^  we  have,  by  substituHn^ 
it  in   equation  (3), 

or,  a;2  =r  4 ; 

whence,  x  z=z2       and       x  =z  —2: 

and  since  y  :=.px   we  have   y  =  1    and    y  =  —  L 

Third  There  are  certain  other  cases  which  admit  of  solution . 
but  for  wnich  no  fixed  rule  can  be  given. 

We  shall  illustrate  the  manner  of  treating  these  cases,  Vj 
the   solution  of  the  following 


"■  to  find  the  values  of  x  and   y. 


176  ELEMENTS  OF  ALGEBRA.*  [CHAP.   VL 


EXAMPLES. 

I.  Given,       -^  =  48, 
/x 

y 

^  =  24, 
^x 

Dividing  the  first  by  the  second,  member  by  member,  we  have 
=r  2,     or    J^  =  2  ;       whence       y  =  4 ; 

and  by  substituting  in   the  second  equation,  we  get 
y^  =  6,       and       x  =  36. 


\/v 


2.  Given,  x  +v^^y  +  y  =    1^>  ) 

„   .  ^         ,     ^       -.  «^   r  to  fmd  the  values  of  x  and  v. 
^^  +     xy  +  y^  =  133,  )  ^ 

Dividing   the   second  by   the   first,    member   by   member,   we 
have 

But,  x+^^/x^  +  yz=zl9: 

adding  these,  member  to  member,  and  dividing  by  2,  we  find 

^  +  y  -=  13, 
which   substituted   in   the  first   equation,  gives, 

J~xy  z=z  6,       or        xy  =:  36,       and        a?  =  — . 

•V  y 

Substituting  this   expression  for  x,  in   the   preceding   equation, 

we  get, 

36 

—  +  y  =  13, 
y 

or,  2^2  _  i3y  ^  _  so  . 


13  /      ^^   .    169       13       5 

whence,  y  ==  ■2"  "  V  ""  ^^  +  "T  =  T  "*"  T 

and  finally,  y  =  ^,       or      y  =  4 ; 

and  since  a;  +  y  =  13, 

a;  =  4,       or       ar  =  9. 


CHAP.   VLJ      EQUATIONS  OF  THE   SECOND  DEGREE.  177 

«5.  Find  the  values  of  x  and  y,  in   the   equations 
ic2  +  3a;  +  y  =  73  —  "^xy 

y2  +  Sj/  +  X=zU. 

By  transposition,  the  first  equation  becomes, 
a;2  +  2ary  +  3a;  +  y  =  73  ; 
t«)   which,   if  the  second  be  added,   member  to  memberj   inere 
results, 

a;2  +  2xy  +  y^ +  4x  +  4y  =  {x  +  yY  +  4{x  +  y)  =  117. 

If,  now,  in  the   equation 

{x  +  yy  +  4{x  +  y)  =  n7, 
we  regard  x  +  y  as  a  single  unknown  quantity,  we  shall  have 


a;  +  y=-2±/ll7  +  4; 
hence,  ^  +  y  =  — 2+11=9, 

and  a;  +  y= -2-11  = -13; 

whence,  x  =  9  —  y,      and      x  =  —  IS  —  y. 

Substituting  these  values  of  x  in  the  second  equation,  we  have 

2/2  +  2y  =  35,     for     x=z  9  —  y, 
and  y^  -\-2y  =  57,     for    a;  =  —  13  —  y. 

The  first  equation  gives, 

y  =  5,      and      y  =  —  7, 
trnd  the  second, 

y= -1+^58;"    and      y=- 1-^58. 
The   corresponding  values  of  x,  are 

a;  r=  4,         a;  =  16  ; 
a;  =  -  12  -^58,       and       a;  =  -  12  +^58. 

4.  Find  the  values  of  x  and  y,   in   the   equations 
a;2y2  +  a;y2  +  xy  =  600  —  (y  +  2)  xh/^ 
x-\-y^  =  14  —  y. 

From  the  first  equation,  we  have 

xY  +  {y''  +  2y)x^y^  +  xy^  +  xy  =  600, 

or,  x^y'{l  +  y^  +  2y)+xy{l+y)    =600, 

or,  agam,      x^y^  (1  +  2/)^  +  ^y  (1  +  y)  =  ^00  ; 

12 


178  ELEMENTS   OF  ALGEBRA.  [CHAP.  VL 

which  is  of  the  form  of  an    equation  of  the   second  degree,  re- 
garding  xy  (1  -f  y)  as  the   unknown  quantity.     Hence, 

:,y  (1  +  y)  =  -  ^  ±^600  +  1  =  -  i  ±  Y^  ; 

and  if  we   discuss  only  the   roots  which  belong  to  the  -f    value 
of  the   radical,  we  have 

^y(i  +  y)  =  ~5  +  Y  =  ^45 

24 

and  hence,  x  =  — ; — - . 

Substituting  this  value  for  x  in  the  second  equation,  we  have 

(y^  +  y?-14(2/2  +  2/)  =  -24; 
whence,  y^  _|_  ^  __  X2,       and       y^  _|_  y  _  2. 

From  the  first  equation,  we  have 

'1     7 

2,=  --±-  =  3,     or     -4; 

and   the   corresponding  values  of  .r,  from   the   equation 

24  _ 

X  =  —- —  =  2. 
y^  +  y 

From   the   second   equation,  we  have 

2/ =  1,       and       y=— 2; 
which  gives  a?  =  12. 

5.  Given,     x^y  +  ^y"^  =  6,     and     x^y^  +  x^y^  =  12,     to  find  the 

(x  =  2 

x^  +  X  +  y  =18  —  y^  )  to   find    the   values   of 
xy  =z    6           )  X  and  y, 

ix  =  S, 

(y  =  2,    or  3;     or    -  3 :+: /S^ 


values  of  x  and  y.  .         ( a;  =  2     or     1, 

Ans.    i 

or     2. 

6.  Given,  _|  -     •  -  •  ^  ~  —      ^   j. 

.  ,  rr  =  3,    or    2  ;     or    —  3  ±  n/3, 


Problems  giving  rise   to  Equations  of  the   Second   Degree  con 
taining  two  or  more  unknown  quantities. 

1.  Find  tw^o  numbers  such,  that  the  sum  of  the  respectiv  . 
products  of  *he  first  multiplied  by  a,  and  the  second  multiplie<j 
by  J,  shall  be  equal  to  2* ;  and  the  product  of  the  one  by 
the  other   equal   to  p. 


CHAP.   VI.]       EQUATIONS  OF  THE   SECOND  DEGREE.  17i) 

Let  X  and  y  denote  the  required  numbers,  and  we  have 
ax  -\-  by  =  2sj 
and  xy  =:p. 


From    the   first 


2.9- 


b     ' 

whcr.ce,   by   substituting  in   the   second,   and   reducing, 
,  ax^  —  2sx  =  —  bp. 

Therefore,  x  =z  —  =b  — V  s^  —  abp. 

'  a         a^  ' 

g        I     

and  consequently,      y  =  — ■  h=  "Ty^^  ""  ^^^' 

Let  a  =  5  =  1  ;   the  values  of  x,   and  y,  then   reduce   to 
X  =  s  ±y/6^2  _^^     and    y  =  s  ^F^s^  — i?; 

whence  w^e  see  that,  under  this  supposition,  the  two  values 
of  X  are  equal  to  those  of  y,  taken  in  an  inverse  order ;  which 
shows,  that  if 

s  -\-^  s^  —  p     represents  the  value  of  x^   s  —^s^  —p 

will   represent  the   co'^ responding  value  of   y,   and  conversely. 

This  relation  i^  ':xplained  by  observing  that,  under  the  last 
supposition,   'h?  /  /en   equations  become 

a;  +  y  =  25,     and     xy  =.p\ 

and  Oie ';,ii':rjtica  is  then  reduced  io  finding  two  numbers  of  which 
the  siir,i  is  2s,  and  their  product  p ;  or  in  other  words,  to  divide 
tt  number  2s,  into  two  such  parts,  that  their  product  may  be  equal 
io  a  given  number  p, 

2,  To  find  four  numbers,  such  that  the  sum  of  the  first  and 
fourth  shall  be  equal  to  2^,  the  sum  of  the  second  and  third 
equal  to  2s',  the  sum  of  their  squares  equal  to  4c2,  and  the 
product  of  the  first  and  fourth  equal  to  the  product  of  tJie 
Kccond   and   third. 


180  ELEMENTS   OF   ALGEBRA*  [CHAP.   VL 

Let  w,  ar,  y,  anl  z,  denote  the  numbers,  respectively.  Then, 
from  the  conditions  of  the   problem,  we   shall  have 

u  +  z  =  2s  1st  condition  ; 

x  +  y  =2s'  2d          " 

u^-i-x^  +  y^  +  z^  =4:c^  3d          " 

.  qiz  =  xy  4th         " 

At  first  sight,  h  may  appear  difficult  to  find  the  values  of 
the  unknown  quantities,  but  by  the  aid  of  an  auxiliary  unknown 
quantity^  they  are  easily  determined.  . 

Let  p  be  the  unknown  product  of  the  1st  and  4th,  or  2d 
and  3d ;   we   shall  then  have 

\z  =.  s 


and 


.ys2  __^, 


J  '  >•  which  give,      \ 

{  uz=p,    )  {. 

\  \  which  give,      -K  ^  

(  xyzzzp^     )  {yz=zs'—Js''^—p. 


Now,  by  substituting  these  values  of  u,  x^  y,  ^,   in   the   third 
equation  of  the   problem,   it  becomes 


and   by   developing   and  reducing, 

4^2  +  45^2  -_  4p  =  4c2  J     hence,     p  =z  s^  -\-  s''^  —  c^. 

Substituting  this  value  for  p,  in  the  expressions  for  u^  x,  y,  z^ 
/fG   find 


[w  =  5+y^c2-5'2,  j  OJ  =  5' +  y^T^"-^, 


U=5-yc2-.'2^  /y=5'-yc2~52. 

These    values    evidently    satisfy     the    last    equation    of    ihe 
problem ;    for 

UZ  =  {S  +y^2— 72)  (5    _yc2-5'2)  =  S2    _  c2  -f  s'^, 


xy  =  {s'i-/^"^^^^)  {s'  -/^^-Z7')  =s'^-c^  +  s  \ 


CHAP.  VI.]   EQUATIONS  OF  THE  SECOND  DEGREE.       18.1 

Remark. — This  problem  shows  how  much  the  introduction 
of  an  unknown  auxiliary  often  facilitates  the  determination  of 
the  principal  unknown  quantities.  There  are  other  problems 
of  the  same  kind,  which  lead  to  equations  of  a  degree  supe- 
rior to  the  second,  and  yet  thej  may  be  resolved  by  the  aid  of 
equations  of  the  first  and  second  degrees,  by  introducing  unknown 
auxiliaries, 

3.  Given  the  sum  of  two  numbers  equal  to  a,  and  the  sum 
of  their   cubes   equal   to   c,    to   find   the   numbers 

ix  +  y    =  a 
By   the   conditions  i 

[x^  -\-  y^  z:z  c. 

Putting     x  z=  s  +  Zj     and     y  =  s  —  z,     we  have     a  =  2s-, 

( a;3  =  s3  +  3^2^  +  35^2  -f-  ^3 
and  < 

ly^  =zs^  —  Ss^z  +  Ssz^  —  z^  : 

hence,   by   addition,  a:^  -f-  y^  =  ^s^    -{-  Gsz^  =z  c ; 


whence,  z^  =  — ,     and     0  =  rb\/- 


2^^ 


65 


~Vg?'   "^^  y  =  ^^\f- 


or,         X  =  8±\/ — ^— -,      and     y  =  s  ::^\  /  —- : 

and  by   substituting  for   5   its   value, 


2      V  \     3a      /        2  ~V       1 


12a 


,  a  lie  —  \a\        a  I  ^.c  ^  d^ 

4.  The  sum  of  the  squares  of  two  numbers  is  expressed  by 
a,  and  the  difference  of  their  squares  by  h :  what  are  the 
numbers?  /^qr^        ITT^l 

^^^•\/~2~'    V  ~2"* 

5.  What  three  numbers  are  they,  which,  multiplied  two  and 
two,  and  each  product  divide  I  by  the  third  number,  give  th€ 
quotients,   a,   6,   c% 

Ans.  .y^l>,    ,/ac,    ^  Ix. 


182  ELEME>'TS  OF  ALGEBRA.        jCHAP.  VI. 

6.  The  sum  of  two  numbers  is  8,  and  the  sum  of  their 
cubes  is  152 :  what   are   the  numbers  ]  Arts,  3  and  5. 

7.  Find  two  numbers,  whose  difference  added  to  the  differ- 
ence of  their  squares  is  150,  and  whose  sum  added  to  the 
sum    of  their   squares,  is   330.  Ans.  9  and   15. 

8.  There  are  two  numbers  whose  difference  is  15,  and  half 
Their  product  is  equal  to  the  cube  of  the  lesser  number :  what 
are   the   numbers  ?  Ans,  S  and  18. 

9.  What  two  numbers  are  those  whose  sum  multiplied  by 
Ihe  greater,  is  equal  to  77;  and  whose  difference,  multiplied 
by  the   lesser,  is   equal   to    121 

Ans,  4  and  7,    or    |  ^  and    ^  ^2, 

10.  Divide  100  into  two  such  parts,  that  the  sum  of  their 
square   roots   may   be    14.  Ans,  64  and  36. 

11.  It  is  required  to  divide  the  number  24  into  two  such 
parts,  that  their  product  may  be  equal  to  35  times  their  differ- 
ence. Ans,  10  and  14. 

12.  What  two  numbers  are  they,  whose  product  is  255,  and 
tbQ    sum  of  whose  squares  is  5141  Ans.  15  and  17. 

13.  There  is  a  number  expressed  by  two  digits,  which,  when 
divided  by  the  sum  of  the  digits,  gives  a  quotient  greater  by 
2  than  the  first  digit ;  but  if  the  digits  be  inverted,  and  the 
resulting  number  be  divided  by  a  number  greater  by  1  than 
the  sum  of  the  digits,  the  quotient  will  exceed  the  former 
quotient   by  2  :    what   is   the   number  ?  A71S,  24. 

14.  A  regiment,  in  garrison,  consisting  of  a  certain  number  of 
companies,  receives  orders  to  send  216  men  on  duty,  each  com- 
pany to  furnish  an  equal  number.  Before  the  order  was  exe- 
cuted, three  of  the  companies  were  sent  on  another  service, 
and  it  was  then  found  that  each  company  that  remained  would 
have  to  send  12  men  additional,  in  order  to  make  up  the  com- 
plement, 216.  How  many  companies  were  in  the  regiment,  and 
what  number  of  men  did  each  of  the  remaining  companies  send 

Ans.  9  companies  :  each  that  remained  sent  36  men. 


CHAP    VI.  J        EQUATIONS  OF  THE   SECOND  •DEGREE.  IvSS 

15.  Find  three  numbers  such,  that  their  sum  shall  be  14,  the 
sum  of  their  squares  equal  to  84,  and  the  product  of  the  first 
and  third  equal  to   the  square   of  the   second. 

Ans,  2,  4  and  8. 

16.  It  is  required  to  find  a  number,  expressed  by  three 
digits,  such,  that  the  sum  of  the  squares  of  the  digits  shall 
be  104 ;  the  square  of  the  middle  digit  to  exceed  twice  the 
product  of  the  other  two  by  4 ;  and  if  594  be  subtracted  from 
the  number,  the  remainder  will  be  expressed  by  ^  the  same 
figures,    but   with     the   extreme   digits   reversed.  Ans.  862. 

17.  A  person  has  three  kinds  of  goods  which  togetner  cost  $230/^. 
A  pound  of  each  article  costs  as  many  -^j  dollars  as  there  are 
pounds  in  that  article :  he  has  one-third  more  of  the  second  than  of 
the  first,  and  3^  times  as  much  of  the  third  as  of  the  second :  How 
many  pounds  has  he  of  each  article  ? 

Ans.  15  of  the  1st,  20  of  the  2d,  70  of  the  3d. 
18.  Two  merchants  each  sold  the  same  kind  of  stuff:  the 
second  sold  3  yards  more  of  it  than  the  first,  and  together, 
they  received  35  dollars.  The  first  said  to  the  second,  "  I 
would  have  received  24  dollars  for  your  stuff."  The  other  re- 
plied, "And  I  would  have  received  12J  dollars  for  yours." 
How  many  yards  did  each  of  then  sell? 


(  1st  merchant     15)  (5 

Ans.       <     ^  ^  >•       or        < 

(2d     -     .     -     18)  J  8. 


19.  A  widow  possessed  13000  dollars,  which  she  divided  into 
two  parts,  and  placed  them  at  interest,  in  such  a  manner,  that 
the  incomes  from  them  were  equal.  If  she  had  put  out  the  first 
portion  at  the  same  rate  as  the  second,  she  would  have  drawn 
for  this  part  360  dollars  interest;  and  if  she  had  placed  the 
second  out  at  the  same  rate  as  the  first,  she  would  have  drawn 
for  it  490  dor.ars  interest.     What  were  the  twc  rates  of  interest  J 

Ans.    7  and  6  per  cent. 


CHAPTER  VII. 

FORMATION     OF    POWERS — BINOMIAL    THEOREM — EXTttACTION    97     ROOTS    OP 
ANY   DEGREE OF    RADICALS. 

128t  The  solution  of  equations  of  the  second  degree  supposes 
the  process  for  extracting  the  square  root  to  be  known.  In 
like  manner,  the  solution  of  equations  of  the  third,  fourth,  &c., 
degrees,  requires  that  we  should  know  how  to  extract  the  third, 
fourth,  &;c.,  roots  of  any   numerical   or   algebraic   quantity. 

The  power  of  a  number  can  be  obtained  by  the  rules  for 
multiplication,  and  this  power  is  subject  to  a  certain  law  of  for- 
mation, which  it  is  necessary  to  know,  in  order  to  deduce  the 
root  from  the  power. 

Now,  the  law  of  formation  of  the  square  of  a  numerical  or 
algebraic  quantity,  is  deduced  from  the  expression  for  the  square 
of  a  binomial  (Art.  47) ;  so  likewise,  the  law  of  a  power  of 
any  degree,  is  deduced  from  the  expression  for  the  same  power 
of  a  binomial.  We  shall  therefore  first  determine  the  law  for 
the  formation  of  any  power  of  a  binomial. 

129.  By  taking  the  binomial  x  -\-  a  several  times,  as  a  factor, 
the  following  results  are  obtained,  by  the  rule  for  multiplicution 

{x  -{-  a)    =  X  -\-  a, 

{x  +  ay  =z  x^  -{-  2ax  -J-  a^^ 

{x  +  a)3  =::x^  +  ^ax^  +  Sa^x   +  a^, 

(X  -{-  ay  z=  x^  +  4ax^  +  Qa^x^  +  4a^x  -f  a\ 

{x  +  ay  =  x^  +  5ax*  +  10^2^3  +  lOa^o^^  +  ^a'^x  +  «*. 

By  examining  these  powers  of  ar  +  «>  we  readily  discover  (h^ 
law   according   to   which   the   exponents  of  the   powers  of  a  lo 


CHAP.   VII.J       PERMUTATIONS  AND   COMBINATIONS.  185 

crease,  and  those  of  the  powers  of  a  increase,  in  the  successive 
terms.  It  is  not,  however,  so  easy  to  discover  a  law  for  the 
formation  of  the  co-efficients.  Newton  discovered  one,  by  means 
of  which  a  binomial  may  be  raised  to  any  power,  without  per 
forming  the  multiplications.  He  did  not,  however,  explain  the 
course  of  reasoning  which  led  him  to  the  discovery ;  but  the  law 
has  since  been  demonstrated  in  a  rigorous  manner.  Of  all  the 
known  demonstrations  of  it,  the  most  elementary  is  that  which 
is  founded  upon  the  theory  of  combinations.  However,  as  the 
demonstration  is  rather  complicated,  we  will,  in  order  to  simplify 
it,  begin  by  demonstrating  some  propositions  relative  to  permu- 
tations and  combinations,  on  which  the  demonstration  of  the 
binomial  theorem  depends. 

Of  Permutations^  Arrangements   ayid    Combinations, 

130»  Let  it  be  proposed  to  determine  the  whole  number  oj 
ways  in  which  several  letters,  a,  ^,  c,  c?,  &;c.,  can  be  written, 
one  after  the  other.  The  result  corresponding  to  each  change 
in  the  position  of  any  one  of  these  letters,  is  called  a  per 
mutation. 

Thus,  the   two   letters   a  and   b  furnish  the  two  permxitations^ 

Jib   and   ba, 

rcah 

acb 

In  like  manner,  the   three   letters,  a,  5,  c,  furnish  abc 

six  permutations.  |  cba 

^       bca 

Permutations,  are  the  results  obtained  by  writing  a  certain 
number  of  letters  one  after  the  other,  in  every  possible  order,  in 
iitich  a  manner  that  all  the  letters  shall  enter  into  each  result^  and 
each  letter  enter  but  once. 

To  determine  the  number  of  permutations  of  which  n  letters  are 
susceptible. 

Two  letters,    a   and  5,    evidently   give    two   per-  j  ab 

mutations.  \ba 


186 


ELEMENTS   OF   ALGEBR^, 


LCHAP.   VI. 


ha 

''cab 

ach 

abe 

cba 

hca 

^  bac 


Therefore,  the  number  of  pemutations  of  two  letters  is  ex 
pressed  by    1x2. 

Take   the    three    letters,   a,    b,   and    c.      Reserve  /  c 

either  of  the   letters,  as   c,  and  permute  the   other 
two,  giving 

Now,  the  third  letter  c  may  be  placed  before  ab, 
between  a  and  5,  and  at  the  right  of  ab ;  and  the 
same  for  ba  :  that  is,  in  one  of  the  first  permuta- 
tions^ the  reserved  letter  c  may  have  three  different 
•places^  giving  three  permutations.  And,  as  the  same 
may  be  shown  for  each  one  of  the  first  permutations, 
it  follows  that  the  whole  number  of  permutations  of 
three  letters  will  be  expressed  by,  1  X  2  X  3. 

If,  now,  a  fourth  letter  d  be  introduced,  it  can  have  four 
places  in  each  one  of  the  six  permutations  of  three  letters  : 
hence,  the  number  of  permutations  of  four  letters  will  be  ex. 
pressed   by,    1  x  2  X  3  x  4. 

In  general,  let  there  be  n  letters,  a,  5,  r,  &c.,  and  suppose 
the  total  number  of  permutations  o^  n  —  \  letters  to  be  known ; 
and  let  Q  denote  that  number.  Now,  in  each  one  of  the  Q  per- 
mutations, the  reserved  letter  may  have  n  places,  giving  n  per- 
mutations :  hence,  when  it  is  so  placed  in  all  of  them,  the 
entire  number  of  permutations  will  be  expressed  by   §  X  n. 

If  n  =z  5,  Q  will  denote  the  number  of  permutations  of  four 
quantities,  or  will  be  equal  to  1x2x3x4;  hence,  the  num- 
ber of  permutations  of  five  quantities  will  be  expressed  by 
1x2x3x4x5. 

IfS^  =  6,  we  shall  have  for  the  number  of  permutations  of 
sk   quantities,  1x2x3x4x5x6,  and   so   on. 

TTence,  if  Y  denote  the  number  of  permutations  of  n  letters, 
"w  -)  shall   have 

F=  Qxnz=zl.     2.     3,     4.     .     .     .     {n^Vjn:     that  is. 

The  number  of  permutations  of  n  letters,  is  equal  to  the  con- 
U-mcd  product  of  the  natural  numbers  from    1    to   n  inclusively. 


CHAP.   VI.]        PEKMUTATIONS  AND  COMBINATIONS.  187 

Arrangements, 

IM.  Suppose  we  have  a  number  m,  of  letters  a,  5,  c,  c?,  &c. 
li/they  are  written  in  sets  of  2  and  2,  or  3  and  3,  or  4  and  4 
.  .  .  in  every  possible  order  in  each  set,  such  results  are  called 
xrrcm^^ents. 

Thus,  ab,  ac,  ad,  .  ,  ,  ba,  be,  bd,  ,  .  .  ca,  cb,  cd,  .  ,  .  are  ar 
^j^ecfTgements  of  m   letters  taken  2  and  2 ;  or  in  sets  of  2  each. 

In  like  manner,  abc,  abd,  .  .  .  bac,  bad,  .  .  .  acb,  acd,  .  .  .  are 
wrangements  taken  In  sets  of  3. 

X  ARRANGEMENTS^«re  the  rcsults  obtained  by  writing  a  number  m 
f  letter&Y^vTsets  of  2  and   2-  3  and  3,  4  and  4,  ,  .  ,  n  and  n ; 
K\e  letters  in    each   set   having    every  possible  order,  and  m  being 
always  greater  than  n. 

If  we  suppose  m  =in,  the  arrangements,  <-aken  n  and  n,  be- 
come permutations. 

Having  given  a  number  m  of  letters  a,  b,  c,  d,  ,  ,  ,  to  deter- 
mine ihe  total  number  of  arrangements  that  may  be  formed  of  them 
by  taking  them  n  in  a  set. 

Let  it  4je  proposed,  in  the  first  place,  to  arrange  three  letters, 
a,    b    and   c,   in   sets   of  two   each. 

First,  arrange  the  letters  in  sets  of  one  each,  and 
for  each  set  so  formed,  there  will  be  two  letters 
reserved:  the  reserved  letters  for  either  arrange- 
ment, being  those  which  do  not  enter  it.  Thus,  with  [^  c 
reference  to  a,  the  reserved  letters  are  b  and  c ;  with  reference 
to  b,  the  reserved  letters  are  a  and  c;  and  with  reference  to  c, 
they   are   a  and   h. 

Now,  to  any  one  of  the   letters,  as   a,   annex,   in  ^^ 

successiDn,   the    reserved    letters   b   and   c :    to    the 
second  arrangement  b,  annex  the   reserved  letters   a  -< 

and  c     and-  to   the   third   arrangement,   c,  annex  the 
reserved  letters   a   and   b. 

Since  each  of  the  first  arrangements  gives  as  many  new 
arrangements   as   there   are   reserved   let*-ers,  it  follows,  that  tht 


ac 
ba 
be 
ca 
cb 


a  c 
ad 
ha 
he 
hd 
ca 
ch 
cd 
da 
dh 
dc 


188  ELEMENTS   OF   ALGEBRA.  [CHAP.   VIL 

number  of  arrangements  of  three  letters  taken^  iwo  in  a  set,  will  be 
equal  to  the  7iumher  of  arrangements  of  the  same  letters  taken  one 
in  a  set^  multiplied  hy  the  number  of  reserved  letters. 

Let  it  be  required  to  form  the  arrangement  of  four  lelterii, 
a,   h^   c   and   d^   taken   three   in   a   set. 

First,  arrange  the  four  letters  in  sets  of  two  ;  there  (ah 

will  then  be  for  each  arrangement,  two  reserved  let- 
ters. Take  one  of  the  sets  and  write  after  it,  in  suc- 
cession, each  of  the  reserved  letters:  we  shall  thus 
form  as  many  sets  of  three  letters  each  as  there  are 
reserved  letters ;  and  these  sets  differ  from  each  other  «    \ 

by  at  least  the  last  letter.  Take  another  of  the  first 
arrangements,  and  annex,  in  succession,  the  reserved 
letters ;  we  shall  again  form  as  many  different  arrange- 
ments as  there  are  reserved  letters.  Do  the  same  for 
all  of  the  first  arrangements,  and  it  is  plain,  that  the 
whole  number  of  arrangements  which  w411  be  formed,  of  four 
letters,  taken  3  and  3,  will  be  equal  to  the  number  of  arrange- 
ments of  the  same  letters^  taken  two  in  a  set,  multiplied  hy  the 
number  of  reserved  letters. 

In  general,  suppose  the  total  number  of  anangements  of  m 
letters,  taken  n  —  \  in  a  set,  to  be  known,  and  denote  this  num- 
ber by  P, 

Take  any  one  of  these  arrangements,  and  annex  to  it,  in  suc- 
cession, each  of  the  reserved  letters,  of  which  the  numbeif  is 
m  —  (7^  -|-  1),  or  m  —  n  -\-  \,  It  is  evident,  that  w^e  shall  thus 
form  a  number  m  —  n  -\-\  of  new  arrangements  of  n  letters, 
each  differing  from  the  others  by  the  last  letter. 

Now,  take  another  of  the  first  arrangements  of  n  —  1  letters, 
and  annex  to  it,  in  succession,  each  of  the  m  —  n  -\-  \  letters 
which  do  not  enter  it ;  we  again  obtain  a  number  m  —  n  -\- \  of 
arrangements  of  n  letters,  differing  from  each  other,  and  from 
those  obtained  as  above,  by  at  least  one  of  the  n  —  \  first  letters. 
Now,  as  we  may  in  the  same  manner,  take  all  the  P  arrange- 
ments of  the  m  letters,  taken  n  ^\  in  a  set,  and  annex  to  them, 


CHAP.   VII.]     PERMUTATIONS  AND   COMBINATIONS.  189 

in  succession,  each  of  the  m  —  n  -{-  1  other  letters,  it  follows 
that  the  total  number  of  arrangements  of  m  letters,  taken  n  ic 
a   set,   is   expressed   by 

F{m  —  n  +  1). 

To  apply  this,  in  determining  the  number  of  arrangements  of 
m  letters,  taken  2  and  2,  3  and  3,  4  and  4,  or  5  and  5  in  a 
set,  make  n  =  2  ;  whence,  m  —  w+l  =  m  —  1;  P  in  this 
case,  will  express  the  total  number  of  arrangements,  taken  2  —  1 
and  2  —  1,  or  1  and  1  ;  and  is  consequently  equal  to  m;  there- 
fore, the  expression 

jP(m  —  n  +  1)   becomes  m[m  —  1). 

Let  71  =  3  ;  whence,  m  —  w+l=m--2;  F  will  then  ex- 
press the  number  of  arrangements  taken  2  and  2,  and  is  equal 
to   m(m  —  1) ;   therefore,  the   expression   becomes 

m{m  —  l){m—  2). 

Again,  take  n  =  4:  whence,  m— w  +  l=m  —  3:  F  will  ex 
press  the  number  of  arrangements  taken  3  and  3,  and  therefore 
the   expression   becomes 

m{m  —  l){m  —  2){m  —  3),     and  so  on. 

Hence,  if  we  denote  the  number  of  arrangements  of  m  let- 
ters, taken   n   in  a  set  by  X,  we   shall   have, 

\, 
X=z  F{m  —  n  +  1)  =m{m  -—I)  (m  —  2)  .  .  (m  — •  w  -f  1)  ;  that  is, 

The  nuinher  of  arrangements  of  m  letters^  taken  n  in  a  set^  u 
equal  to  the  continued  product  of  the  natural  numbers  from  m 
down  to   m  —  n  +  1,  inclusively/. 

If  in  the  preceding  formula  m  be  made  equal  to  n,  the  ar 
rangements  become  permutations,  and  the  formula  reduces  to 

X=n{n^l){n^2)  .  ...2.1; 

•  /r,  by  reversing  the  order  of  the  factors,  and  writing   Y  for  X, 

r=l\  2  .  3  .   .   .   .   (7i-l)/i; 

the   ,5ame  formula  as  deduced  in   the  last  arti(5le. 


190  ELEMENTS  OF  ALGEBRA.  [CHAP.   VTT 

Combinations, 

132i  When  the  letters  are  disposed,  as  in  the  arrangements, 
2  and  2,  3  and  3,  4  and  4,  &c.,  and  it  is  required  that  any 
two  of  the  results,  thus  formed,  shall  differ  by  ^t  least  one 
letter,  the  products  of  the  letters  will  be  different.  In  this  case, 
the  results   are   called   combinations. 

Thus,  mb^  ac^  be,  ^.  .  .  ad,  bd,  ,  ,  ,  are  combinations  of  the  let 
ters  a,  5,  c,  and  d,  &;c.,  taken   2   and   2. 

In  like  manner,  aba,  abd,  .  .  .  acd,  bed,  .  .  .  are  combinatioms 
of  the  letters   taken   3   and   3 :    hence. 

Combinations,  are  arrangements  in  which  any  two  will  differ 
from  each  other  by  at  least  one   of  the  letters  which  enter  them. 

To  determine  the  total  number  of  different  combinations  thai 
can  be  formed  of  m   letters,   taken   n  in  a   set. 

Let  X  denote  the  total  number  of  arrangements  that  can  be 
formed  of  m  letters,  taken  n  and  n ;  Y  the  number  of  per 
mutations  of  n  letters,  and  Z  the  total  number  of  different 
combinations   talcen   n   and   n. 

It  is  evident,  that  all  the  possible  arrangements  of  m  letters 
taken  n  in  a  set,  can  be  obtained,  by  subjecting  the  n  letters 
of  each  of  the  Z  combinations,  to  all  the  permutations  of  which 
these  letters  are  susceptible.  Now,  a  single  combination  of  n 
letters  gives,  by  hypothesis,  Y  permutations  or  arrangements  • 
therefore  Z  combinations  will  give  Y  X  Z  arrangements ;  and 
as   X  denotes  the  total  number  of  arrangements,  it  follows  ihat 

X=  Yx  Z:      whence,      Zz:^y, 

But   we  have  (Art.  130), 

Y-Qxnz=zl,2.^,,,,n, 
and  (Art.  131), 
X^F(m-n^-\)^m{m-l)(m-2)  ,  .  .  .  (tti  ~  w  f  1) ; 
therefore, 
__  P  (w  —  n  4-  1)  _  m  {m  —  1)  (m  —  2)  .  .  .  .  {m—  n-^l)  ^ 

~         Qx'n         "~  1.2.3 n 

that  is. 


CHAP.   VII.J  BINOMIAL  THEOREM.  191 

The  number  cf  comhinations  of  m  letters  taken  n  in  a  set^ 
is  equal  to  the  continued  product  of  the  natural  numbers  from 
m  down  to  m  —  n  -{-  1  inclusively^  divided  by  the  continued 
product   of  the   natural   numbers  from   1    to   n   inclusively. 

133.  If  Z  denote  the  number  of  combinations  of  the  m  let- 
ters  taken   n   in   a   set,  we   have  just   seen   that 

m{m-l){m-2)  .  .  .  .  {m  -  n  +  1) 
^  = 1.2.3 n  ^^^- 

If  Z'  denote  the  number  of  combinations  of  m  letters  taken 
(m  —  n)  in  a  set,  we  can  find  an  expression  for  Z'  by  chang- 
ms  n  into  m  —  n  in  the  second  member  of  the  above  formula ; 
whence 

_  m(m-l)(m-2) {n -\-  \) 

1.2.3 (m  -  n)        ^  ^' 

If,  now,  we  divide  equation  (1)  bj  (2),  member  by  member, 
and  arrange  the  factors  of  both  terms  of  the  quotient,  we 
shall  have 

Z    __  1  .  2  .  3  .  .  .  .  (m  --  n)  X  (m  —  n  +  1)    .    .    .    {m  —  l)m 

Y'  ""  1.2.3....  .      7^  X  (/^  +  1) {m  —  l)m' 

The  numerator  and  denominator  of  the  second  member  are 
equal  to  each  other,  since  each  contains  the  factors,  1,  2,  3, 
&c.,  to  m;   hence, 

^  =  1,     or     Z  =.  Z' \     therefore, 

Li 

The  number  of  combinations  of  m  letters^  taken  n  in  a  set,  is 
equal  to  the  number  of  combinations  of  m  letters^  taken  m  —  n  in 
a  set. 

Binomial  Theorem. 

134.  The  object  of  this  theorem  is  to  show  how  to  ficd  any 
power  of  a  binomial,  without  going  through  the  process  of  con 
turned  multiplication. 

135.  The  algebraic  equation  which  indicates  the  law  of  for- 
mation of  any  power  of  a  biromial,  is  called  the  Binomicu 
Formula, 


192 


ELEMENTS   OF  ALGEBRA. 


[CHAP.   Vll. 


In  order  to  discover  this  law  for  the  mth  power  of  the  bino- 
mial X  -\-  a,  let  us  observe  the  law  for  the  formation  of  the 
product  of  several  binomial   factors,   x  -{-  a,   x  -\-  b,   x  -{-  c,   x  -\-d 

.  .  of  which  the  first  term  is  the  same  in  all,  and  the  second 
terms   different. 


1st  product 


2d 


x    +  a 
X    -\-  b 


x^  +  a 
a;    +  c 


X    -{-  ab 


a:3  +  a 

a;2  +  ab 

+  6 

+  ac 

+  c 

+  5c 

X    +  abc 


3d 


ic*  +  a 

x^  +  oh 

a;2  4-  abc 

+  6 

+  ac 

+  abd 

+  c 

+  ad 

+  ace? 

+  c? 

+  be 
-f  bd 
+  cc? 

+  bed 

X     +  ^<^Gf 


These  products,  obtained  by  the  common  rule  for  algebraic 
multiplication,   indicate   the   following   laws : — 

1st.  With  respect  to  the  exponents,  we  observe  that  the  ex- 
ponent of  X,  in  the  first  term,  is  equal  to  the  number  of  bino- 
mial factors  employed.  In  each  of  the  following  terms  to  the 
right,,  this  exponent  is  diminished  by  1  to  the  last  term,  where 
it   is   0. 

2d.  With  respect   to   the   co-efficients   of  the   different  powers 

of  X,  that  of  the   first  term   is   1 ;  the  co-efficient  of  the  second 

""term  is  equal  to  the  sum  of  the  second  terms  of  the  binomials ; 

the   co-efficient   of  the   third   term   is   equal   to   the   sum  of  the 

products  of   the    different    second  terms,   taken  two   and  two; 


CHAP.   VII.]  BINOMIAL   THEOBEM.  193 

the  co-efRcient  of  the  fourth  term  is  equal  to  the  sum  of  their 
different  products,  taken   three   and   three. 

Eeasoning  from  analogy^  we  might  conclude  that,  in  the  pro- 
duct of  any  number  of  binomial  factors,  the  co-efficient  of  the 
term  which  has  n  terms  before  it,  is  equal  to  the  sum  of  the 
different  products  of  the  second  terms  of  the  binomials,  taken 
n  and  n.  The  last  term  of  the  product  is  equal  to  the  con- 
tinued  product  of  the   second   terms  of  the   binomials. 

In  order  to  prove  that  this  law  of  formation  is  general,  sup- 
pose that  it  has  been  proved  true  for  the  product  of  m  bino- 
mials. Let  us  see  if  it  will  continue  to  be  true  when  the 
product  is  multiplied  by  a  new  binomial  factor  of  the  same 
form. 

For   this   purpose,   suppose  . 

to  be  the  product  of  m  binomial  factors;  iVic^-"  repiesenting  the 
term  which  has  n  terms  before  it,  and  Mx'^"^'^^  the  term  which 
immediately  precedes. 

Let  X  +  k  ho.  the  new  binomial  factor  by  which  we  multiply ; 
the  product,  when  arranged  according  to  the  powers  of  a:, 
will  be 


+  k\       +  Ak 


+  Bk 


'  +  ...  +Jsr 

-\-Mk 


from  which  we  perceive  that  the  law  of  the  exponents  is  evi- 
dently the  same. 

With  respect  to  the  co-efficients,  we  observe; 

1st.  That  the  co-efficient  of  the  first  term  is  1 ;  and 

2d.  That  A-\-  k^  or  the  co-efficient  of  a;"*,  is  the  sum  of  the 
second  terms  of  the  m  -\-  1  binomials. 

3d.  Since,  by  hypothesis,  B  is  the  sum  of  the  different  products 
of  the  second  terms  of  the  m  binomials,  taken  two  and  two,  and 
since  A  X  k  expresses  the  sum  of  the  products  of  each  of  the 
second  terms  of  the  first  m  binomials  by  the  new  second  term  k ; 
therefore,  B  -\-  Ak  is  the  sum  of  the  different  products  of  tlie 
second  terms  of  the  m  -}-  1  binomials,  taken  two  and  two. 


194  ELEMENTS   OF  .  ALGEBlti.  [CHAP.   VIT. 

In  general,  since  N  expresses  the  sum  of  the  products  of  the 
Becond  terms  of  the  m  binomials,  taken  n  and  n^  and  M  the  sum 
of  their  products,  taken  ti  —  1  and  7^  —  1,  therefore  N -^  Mk^ 
or  the  co-efficient  of  the  term  which  has  n  terms  before  it,  will  be 
equal  to  the  sum  of  the  diiferent  products  of  the  second  teriua 
of  the  77^  +  1  binomials,  taken  n  and  7i.  The  last  term  \9 
equal  to  the  continued  product  of  the  second  terms  of  the  m  -f  1 
binomials.  * 

Hence,  the  law  of  composition,  supposed  true  for  a  number  m 
of  binomial  factors,  is  also  true  for  a  number  denoted  by  m  +  \, 

But  we  have  shown  the  law  of  composition  for  4  factors, 
hence,  the  same  law  is  true  for  5 ;  and  being  true  for  5,  it 
must  be  for  6,  and  so  on;    hence,  it  is  general. 

136.  Let  us   take   the   equation, 

(x-[-a){x  +  h){x-^c)   .   ...    =  a;'"  +  Ax"^^  +  Bx"^^      .... 
_|_  JSfx^ri  .   .   .   .    +  TT, 

containing  in  the  first  member,  m  binomial  factors.     If  we  make 

ar=6=:Cr=C?-.   .   .   .   (Sec, 
the   first  member   becomes, 

{x  +  a)^. 
In   the   second   member   the   co-efficient  of  x'^  will   still   be  1. 
The   co-efficient  of  ir^-\  being   a  +  6  +  c  +  c?,  .  .  .  will  become 
a  taken   m   times ;    that  is,    ma. 
The   co-efficient  of  ir"*"^^  being 

ah  +  ac  -\-  ad  ,  .  .  .  reduces   to  a^  +  a^  +  a^  ,  ,  , 
that  is,  it  becomes   a^   taken  as  many  times  as  there  are  com 
binations   of  m  letters,  taken  two  and   two,  and   hence  reduces 

(Art.  132),  to 

m  —  1    „ 


2 

The   co-efficient   of   a;''*"^   reduces   to   the    product   of   a^,    multi- 
plied by   the    number   of   different    combinations   of    m   letters 
taken   three   and   three ;   that   is,  to 
m  —  1   m  —  2 


2      •       3 


a^,  (tec. 


CHAP.   VIIJ  BINOMIAL  THEOREM.  195 

Let  us  denote  the  general  term,  that  is^  Me  one  which  has 
n   terms   before   it,   by   iVa:*^". 

Then,  the  co-efficient  iV  will  denote  the  sum  of  the  products 
of  the  second  terms,  taken  n  and  n  ;  and  when  all  tlie 
second  terms  are  supposed  equal,  it  becomes  equal  to  a"  mul- 
tiplied by  the  number  of  combinations  of  m  letters,  taken 
n  and  n.  Therefore,  the  co-efficient  of  the  general  term  (Art. 
132),  is 

Q  Xn  '  ^ 

hence,  we   have,  by  making   these   substitutions, 

^  —  1    « 


{x  +  a)^  =  x^  +  maoif^^  + 


m. 


2 


m  —  \   m  —  2^       ^  ,  F(m  —  n-}-!) 

+  m, — ^r — . — - — a^x"^-^  »  .  .  +         ^ ia^'x'"^  ...    +  a* 

2  6  V  •  ^ 

which  is  the  binomial  formula. 

The  term 

F(m  —  n-{-l) 

— !^ 1 — i  a^x^""-^ 

Qn 

is   called  the  general  term,  because   by  makmg   w  =  2,  3,  4,  &c., 

all   the   others   can  be   deduced  from  it.      The  term  which   im 

mediately  precedes  it,  is 

F  F 

^n-i<j,m-«  +  i^     since    — 

evidently   expresses  the    number   of    combinations   of  m  letters 
taken   n -— 1    and  n  —  1.     Hence,  we   see,  that 
F{m  —  n+  1) 

which  is   called   the  numerical   co-efficient  of  the   general   term, 

p 

is  equal  to  the  numerical  co-efficient    —    of  the  preceding  term, 

multiplied   by  m  ^  n  +  I,  the   exponent  of  x  in   that  term,  and 
divided  by  n,  the  number  of  terms  preceding  the  required  term. 

The  simple  law,  demonstrated  above,  enables  us  to  determine 
the  numerical  co-efficient  of  any  term  from  that  of  the  preceding 
term,  by  means  of  the  following 


196  ELEMENTS   OF  ALGEBRA.  [CHAP.   VIL 

RULE. 

The  numerical  co-efficient  of  any  term  after  the  first,  is  forifiud 
hy  multiplying  that  of  the  preceding  term  by  the  exponent  of 
T  in  that  term,  and  dividing  the  product  hy  the  number  of 
terms   which  precede    the   required    term. 

137«  Let  it  be   required   to    develop 

{x  +  ay. 

By  applying   the  foregoing  principles,  we  find, 
{x  I-  ay  =zx^-{-  6ax^  +  Iba'^x^  +  20a^x^  +  15tt%2  +  5^5^  _f_  ^6^ 
Having  written  the  first  term   a;^,  and  the  literal  parts  of  the 
oiher   terms,    we   find   the   numerical   co-efficient    of   the    second 
term   by   multiplying   1,   the   numerical    co-efficient   of   the   first 
term,  by  6,  the   exponent   of  x    in   that  term,  and  dividing   by 
1,  the  number  of  terms  preceding  the  required  term.     To  obtain 
the   co-efficient   of  the   third   term,  multiply  6    by  5    and  divide 
the   product  by  2 ;  we   get   15  for   the  required   number.      The 
other  numerical  co-efficients  may  be  found  in  the  same  manner 
In  like  manner,  we  find 

{x  +  ay^  =  x^^  -f  lOax^  +  A^a^x^  +  120aV  +  2l0a^x^ 
+  252a^x^  +  210aV  -f  I20a'^x^  +  45a%2  _|_  lo^Q^  -f.  a^^. 

138*  The  operation  of  finding  the  numerical  co-efficients  may 
be   much  simplified  by   the   aid   of  the   following  principle. 

We  have  seen  that  the  development  of  (x  -f-  a)^,  contains 
m  +  1  terms ;  consequently,  the  term  which  has  n  terms  afler 
it,  has  m  —  n  terms  before  it.  Now,  the  numerical  co-efficient 
of  the  term  which  has  n  terms  before  it  is  equal  to  the  num> 
ber  of  combinations  of  m  letters  taken  ti  in  a  set,  and  the 
numerical  co-efficient  of  that  term  which  has  n  terms  after  it, 
01  m  —  n  before  it,  is  equal  to  the  number  of  combinations  of 
m  letters  taken  m  —  n  in  a  set ;  but  we  have  shown  (Art.  133) 
that   these   numbers   are   equal.     Hence, 

In   the    development   of  any  power   of  a  binomial  of  the  form 
\x  +  ay,  the  numerical  co-efficients  of  terms  at  equal  distances  from 
the  two  extremes,  are  equal  to  each  other. 


CHAP.   VII.]  BINOMIAL  THEOREM.  197 

We  see  that  this  is  the  case  in  both  of  the  exairples  above 
given.  In  finding  the  development  of  anj  power  of  a  binomial, 
we  need  find  but  half,  or  one  more  than  half,  of  the  numerical 
co-efficients,  since  the  remaining  ones  may  be  written  directly 
from  those  already  found. 

139.  It  frequently  happens  that  the  terms  of  the  binomial, 
to  which  the  formula  is  to  be  applied,  contain  co-efficlenta 
and   exponents,    as   in   the   following   example. 

Let   it   be   required   to   raise   the   binomial 
Sa'^c  —  2bd 
to   the   fourth   power. 

Placing  Sa^c  =  x      and       ~  2bd  =  y,       we  have 

{x  +  7/Y  =:  X^  +  4:X'^y  +  6a;2y2  _|_  4^y3  +  y*  ; 

and   substituting  for  x   and  y   their   values,  we   have 

(3a2c  --  2hdY  =  {Sa^cY  +  4  {Sa^cY  (  -  2bd)  +  6  {Sa-'cY  (-  2bdY 

+  4  {Sa^c)  (-  2bdY  +  (-  2bdY, 
or,  by  performing   the   operations   indicated, 

(Sa^c  -  2bdY  =  Sla^c^  -  216a^c^d  +  21Qa^c^W  -  96a^cPd^ 
+  16b^d\ 

The  terms  of  the  development  are  alternately  plus  and 
minus,  as  they  should   be,  since   the    second   term   is    — . 

140.  A  power  of  any  polynomial  may  easily  be  found  by 
means   of  the  binomial   formula,  as   in   the   following  example. 

Let  it  be   required   to   find   the   third  power   of 

a  +  6  +  c. 
First,  put  b  -\-  c  =r  d. 

Then     {a  +  b  +  cY  =  {a  -]-  dY  =  a^  +  ^a?d  +  Zad'^  +  d\ 
and  by  substituting  for  the  value  of  c?, 

(a  +  6  +  c)3  =  a3  +  3a26  +  3a52  +  6' 

Za?c  +  W^c  +  Qabc 
+  3ac2  -f  36^2 


198  ELEMENTS   OF   ALGEBK^.  [CHAP.    VIL 

This  developmei  t  is  composed  of  the  sum  of  the  etches  of  the 
three  terms,  plus  the  sum  of  the  results  obtained  by  multiphjing 
three  times  the  square  of  each  terra,  by  each  of  the  other  terms  in 
succession,  plus  six  times  the  product  of  the  three  terms. 

To  apply  the  preceding  formula  to  the  development  of  the 
cube  of  a  trinomial,  in  which  the  terms  are  affected  with  co- 
efficients and  exponents,  designate  each  term  by  a  single  letter^ 
and  perform  the  operations  indicated ;  then  replace  the  letters 
introduced,  by    their   values, 

From   this   rule,  we  find    that 

(2a2  -  4.ab  +  362)3  ^  Sa^  _  ^^a^b  +  \^2a^b'^  -  2ma?h^ 
+  198a26*  -  108aZ>5  _|.  21b\ 
The  fourth,  fifth,  &c.,  powers   of  any   polynomial   can  be   de- 
veloped in  a  similar   manner. 

Extraction   of  the   Cuhe  Root  of  Numbers, 

141  •  The  cube  root  of  a  number,  is  such  a  number  as  being 
taken   three   times  as  a  factor,  will  produce  the  given  number. 

A  number  whose  cube  root  can  be  exactly  found,  is  called  a 
perfect  cube ;  all  other  numbers  are  imperfect  cubes. 

The  first   ten   numbers   are, 
1,     2,       3,       4,         5,         6,         7,         8,         9,         10; 

and  their  cubes, 
1,     8,     27,     64,      125,     216,     343,     512,     729,     1000. 

Conversely,  the  numbers  in  the  first  line  are  the  cube  roots 
of  the  corresponding  numbers  in   the  second. 

If  we  wish  to  find  the  cube  root  of  any  number  less  than 
1000,  we  look  for  the  number  in  the  second  line,  and  if  't  is 
there  written,  the  corresponding  number  in  the  first  line  will  be 
its  cube  root.  If  tl>3  number  is  not  there  written,  it  will  fall 
between  two  numbers  in  the  second  line,  and  its  cube  root 
will  fall  between  the  corresponding  numbers  in  the  first  line, 
in  this  case  the  cube  root  cannot  be  expressed  in  exact  parts 
of  1 ;  hence,  the  given  number  must  be  an  imperfect  cube  (R€> 
mark  III,  Art.  95). 


CHAP.   Vll.j 


C'JBE  ROOT  OF  NUMBERS. 


199 


If  the  given  number  is  greater  than  1000,  its  cube  root  will 
be  greater  than  10  ;  that  is,  it  will  contain  a  certain  number 
of  tens  and  a   certain  number  of  units. 

Let  us  designate  any  number  by  iV,  and  denote  its  tens  by 
a,  and  its  units  by  h ;  we  shall   have, 

N=a-{-h)       whence,      N^  =  a^ -\-  Za?h  +  Zah'^  +  6^ ;       that  is, 

The  cube  of  a  number  is  equal  to  the  cube  of  the  tens,  plus   three 
times  the  product  of  the  square  of  the  tens  by  the  units,  plus  three 
times  the  product   of  the  tens  by  the  square  of  the  units,  plus  the 
cube  of  the  units. 
Thus  (47)3= (4Q>'  +  3  X  (40)2  x  7  +  3  X  40  X  (7)2  +  (7)^  =,  103823. 

Let  us  now  reverse  the  operation,  and  find  the  cube  root  of 
103823. 


103  823 
64 


42  X  3  =  48  I  398^23 


47 
~8 


48 

47 

48 

47 

384 

329 

192 

188 

2304 

2209 

48 

47 

18432 

15463 

9216 

8836 

110592 


103823 


Since  the  number  is  greater  than  1000,  its  root  will  contain 
tens  and  units.  We  will  first  find  the  number  of  tens  in  the 
root.  Now  the  cube  of  tens,  giving  at  least  thousands,  we  point 
off  three  places  of  fig  ires  on  the  right,  and  the  cube  of  the  num- 
ber of  tens  will  be  f  >und  in  the  number  103,  to  the  left  of  this 
pel  iod. 

The  cube  root  of  the  greatest  cube  contained  in  103  being  4, 
this  is  the  number  of  tens  in  the  required  root.  Indeed,  103823 
is  evidently  comprised  between  (40)^  or  64,000,  and  (50)^  or 
125,000 ;  hence,  the  required  root  is  comprised  between  4  tens 
and  5  tens:  that  is,  it  is  composed  of  4  tens,  plus  a  certain 
number  of  un'.ts  less  than  ten. 


200  ELEMENTS   OF  ALGEBftA.  LCHAP.   VII. 

Having  found  the  number  of  tens,  subtract  its  cube,  64,  fron\ 
103,  and  there  remains  39,  to  which  bring  down  the  part  823, 
and  we  have  39823,  which  contains  three  times  the -product  of 
the  square  of  the  tens  by  the  utiits,  plus  three  times  the  product 
of  the  tens  by   the  square  of  the  units,  plus  the  cube  of  the  units. 

Now,  as  the  square  of  tens  gives  at  least  hundreds,  it  follows 
that  the  product  of  three  times  the  square  of  the  tens  by  the 
units,  must  be  found  in  the  part  398,  to  the  left  of  23,  which 
is  separated  from  it  by  a  dash.  Therefore,  dividing  398  by  48, 
which  is  three  times  the  square  of  the  tens,  the  quotient  8  will 
be  the  units  of  the  root,  or  something  greater,  since  398  is 
composed  of  three  times  the  squarp  of  the  tens  by  the  units,  and 
generally  contains  numbers  coming  from  the   two  other  parts. 

We  may  ascertain  whether  the  figure  8  is  too  great,  by  form- 
ing from  the  4  tens  and  8  units,  the  three  parts  which  enter  into 
39823  ;  but  it  is  nmch  easier  to  cube  48,  as  has  bee.\i  done  in 
the  above  table.  Now,  the  cube  of  48  is  110592,  which  is 
greater  than  103823;  therefore,  8  is  too  great.  By  cab:ng  47, 
we  obtain  103823 ;  hence  the  proposed  number  is  a  ]-erfeot  cube, 
and  47  is  its  cube  root. 

By  a  course  of  reasoning  entirely  analogous .  to  that  pvrsucd 
in  treating  of  the  extraction  of  the  square  root,  we  may  shew 
that,  when  the  given  number  is  expressed  by  more  than  six 
figures,  we  must  point  off  the  number  into  periods  of  three  figui<5S 
each,  commencing  at  the  right.  Hence,  for  the  extraction  of  the 
cube  root  of  numbers,  we  have  the  following 

RULE 

I.   Separate   the  given  number  into  p>eriods  of  three  figures  cacK 
leginning  at  the   right  hand ;  the  left  hand  period   will  often  con 
tain  less  than   three  places  of  figures. 

IL  Seek  the  greatest  perfect  cube  in  the  first  period,  on  the  left, 
and  set  its  root  on  the  right,  after  the  manner  of  a  quotient  ip 
division.  Subtract  the  cube  of  this  nwnber  from  the  first  period, 
and  to  the  remainder  bring  down  the  first  figure  of  the  next  period, 
and  call  this  number  the  dividend. 


(1HAP.    VII.J  EXTKACTION   OF   ROOTS.  201 

III.  Take  three  times  the  square  of  the  root  just  found  for  a 
divisor^  and  see  how  often  it  is  contained  in  the  dividend,  and 
place  the  quotient  for  a  second  figure  of  the  root.  Then  cube  the 
number  thus  found,  and  if  its  ciibe  be  greater  than  the  first  two 
periods  of  the  given  number^  diminish  the  last  figure  by  1 ;  but 
if  it  be  less,  subtract  it  from  the  first  two  periods,  and  to  the 
remainder  bring  down  the  first  figure  of  the  next  period,  for  a  new 
dividend. 

IV.  Take  three  times  the  square  of  the  whole  root  for  a  new 
divisor,  and  seek  how  often  it  is  contained  in  the  new  dividend ; 
the  quotient  will  be  the  third  figure  of  the  root.  Cube  the  number 
thus  found,  and  subtract  the  result  from  the  first  three  periods 
of  the  given  number,  and  proceed  in  a  similar  way  for  all  the 
periods. 

If  there  is  no  remainder,  the  number  is  a  perfect  cube,  and  the 
root  is  exact :  if  there  is  a  remainder,  the  number  is  an  imper- 
fect cube,  and  the  root  is  exact  to  within  less  than  1. 

EXAMPLES. 

1.  3/48228544  Ans.  3G4. 

2.  ^27054036008  Ans.  3002. 

3.  3^483249  Ans.  78,  with  a  remainder  8697. 

4.  3/91632508641         Ans.  4508,  with  a  remainder  20644129. 

5.  y  32977340218432  Ans.  32068. 

Extraction  of  the   N*^   Boot  of  Numbers, 

142«  The  n^^  root  of  a  number  is  such  a  number  as  being 
taken  n  times  as  a  factor  will  produce  the  given  number,  n  being 
%nj  positive  whole  number.  When  such  a  root  can  be  exactly 
found,  the  given  number  is  a  perfect  n*^  power;  all  other  num- 
bers are  imperfect  n*^  powers. 

Let  iV  denote  anj  number  whatever.  If  it  is  expressed  by 
less  than  n  -}-  1  figures,  and  is  a  perfect  n^^  power,  its  n^^  root 
will   be   expressed  by   a   single  ^gure,    and   may   be   found   by 


202  ELEMENTS   OF  ALGEBRA.  [CHAP.   "VXI 

means  of  a  tab\3  containing  the  n^^  powers  of  the  first  ten 
numhers. 

If  the  number  is  not  a  perfect  n^^  power,  it  will  fall  between 
two  V'^^  powers  in  the  table,  and  its  root  will  fall  between  the 
n*^  roots  of  these   powers. 

If  the  given  number  is  expressed  by  more  than  n  figures, 
its  root  will  consist  of  a  certain  number  of  tens  and  a  certain 
number  of  units.  If  we  designate  the  tens  of  the  root  by  a, 
and   the   units   by  6,  we  shall   have,  by  the  binomial  formula, 

]Sf=z{a  +  hY  ^a""  +  na^'-^h  +  n^^—  a^'-W  +,  &c. ; 

that  is,  the  proposed  number  is  equal  to  the  n*^  power  of  the 
tens,  plus  n  times  the  product  of  the  n  —  \^^  power  of  the  tens 
by  the  units,  plus  other  parts  which  it  is  not  necessary  to 
consider. 

Now,  as  the  n*^  power  of  the  tens,  cannot  be  less  than 
1  followed  by  n  ciphers,  the  last  n  figures  on  the  right,  cannot 
make  a  part  of  it.  They  must  then  be  pointed  off,  and  the  n^^ 
root  of  the  greatest  n^^  power  in  the  number  on  the  left  will 
be   the  number  of  tens  of  the   required   root. 

Subtract  the  n*^  power  of  the  number  of  tens  from  the  num 
ber  on  the  left,  and  to  the  remainder  bring  down  one  figure  of 
the  next  period  on  the  right.  If  we  consider  the  number  thus 
fuund  as  a  dividend,  and  take  n  times  the  {n  —  l)^'^  power 
of  the  number  of  tens,  as  a  divisor,  the  quotient  will  evidently 
be   the   number   of  units,  or  a  greater   number. 

If  the  part  on  the  left  should  contain  more  than  n  figures,  the 
n  figures  on  the  right  of  it,  must  be  separated  from  the  rest, 
and  the  root  of  the  greatest  n^^  power  contained  in  the  part 
on  the  left   extracted,  and   so   on.     Hence  the  following 

RULE. 

I.  Separate  the   namher  iV  into  periods   of  n  figures   each,    he 
ginning  at    the   right  hxnd ;    extract  the   n^^   root   of    the  greatest 
perfect   n*^  power  contained  in  the  left  hand  period^  it  will  he  th$ 
first  figure  of  the  root. 


CHAP.   VII.]  EXTRACTIOIS'   OF   ROOTS.  203 

II.  Subtract  this  n*^  power  from  the  left  hand  period  and  bring 
down  to  the  right  of  the  remainder  the  first  figure  of  the  nexi 
period,  and  call  this  the  dividend, 

[II.  Form  the  n  —  1  power  of  the  first  figure  of  the  root,  mul- 
tiply it  by  n,  and  see  how  often  the  product  is  contained  in  the 
dividend:  the  quotient  will  be  the  second  figure  of  the  root,  or 
something  greater, 

IV.  Raise  the  number  thus  formed  to  the  n*^  power,  then  sub- 
tract this  result  from  the  two  left-hand  periods,  and  to  the  new 
remainder  bring  down  the  first  figure  of  the  next  period  :  then 
divide  the  number  thus  formed  by  n  times  the  n  —  1  power  of 
the  two  figures  of  the  root  already  found,  and  continue  this  opera- 
Hon   until   all  the  periods  are  brought   down, 

EXAMPLES. 

1.  What  is  the   fourth   root  of  531441? 
53  1441  I  27 

4  X  23  =  32  I  371 
(27)4=  531441. 

We  first  point  off,  from  the  right  hand,  the  period  of  four 
figures,  and  then  find  the  greatest  fourth  root  contained  in  53, 
the  first  period  to  the  left,  which  is  2.  We  next  subtract  the 
4th  power  of  2,  which  is  16,  from  53,  and  to  the  remainder 
37  we  bring  down  the  first  figure  of  the  next  period.  We 
then  divide  871  by  4  times  the  cube  of  2,  which  gives  11  for 
a  quotient :  but  this  we  know  is  too  large.  By  trying  the  num- 
bers 9  and  8,  we  find  them  also  too  large :  then  trying  7,  we 
find  the  exact  root  to  be  27. 

143.  When  the  index  of  the  root  to  be  extracted  is  a  multiple 
of  two  or  more  numbers,  as  4,  6,  .  .  .  &;c.,  the  root  can  be  ob- 
tained by  extracting  roots  of  more  simple  degrees,  successively.  To 
explain  this,  we  will  remark  that, 

{ci^y  =  a3  X  a3  X  a3  X  a^  =  a3  +  3  +  3  +  3  _  ^sy*  —  a^a^ 
and,  in  general,  from  the  definition  of  an  exponent 
(a'^y  =-.  a^  >  a"^  X  a""  X  a"^  ,  ,  ,   =  a"»X» : 


204  ELEMENTS   OF  ALGEBRA.  [CHAP.   VIL 

hence,  the  n^^  power  of  the  m*'^  power  cf  a  number  is  equal  to  thi 
fji^th  power  of  this  number. 

Let  us  see  if  the   converse  of  this  is  alsc  true. 


Let 

then  raising  both  members  to  the  n*^  power,  we  have,  from  the 
definition  of  the   n^^  root, 

y^=b^', 
and  by  raising  both  members  of  the  last  equation  to  the  m*^  power 

a  =  6'"'*. 
Extracting  the  mn*^  root  of  both  members  of  the  last  equation, 

mn  Jq^  "k  . 

we  have,  y  ^  —  ^  ? 


and  hence,  \J '^J~a  ^=z  "^^Ta^ 

since  each  is  equal  to  h.  Therefore,  the  n^^  root  ofjhe  rpJ^*  root 
of  any  number^  is  equal  to  the  mn^^  root  of  that  number.  And 
in  a  similar  manner,  it  might  be  proved  that 


By  this  method  we  find  that 

2.  y  2985984  =  W^  2985984  =  ^/l728  =  12. 

3.  6^1771561  =  J ^  1771561  =  11. 

4.  8/1679616  =  yi296=::y^.^/l296==G. 

Eemark. — Although  the  successive  roots  may  be  extracted  in 
any  order  whatever,  it  is  better  to  extract  the  roots  of  the  lowest 
degree  first,  for  then  the  extraction  of  the  roots  of  the  higher 
degrees,  which  is  a  more  complicated  operation,  is  effected  upon 
numbers  containing  fewer  figures  ^han  the  proposed  number. 


CHAP.   yil.j  EXTRACTION  OF  ROOTS.  205 

Extraction  of  Boots  hy  Approximation, 

144.  When  it  is  required  to  extract  the  n*^  root  of  a  number 
which  is  not  b,  perfect  n^^  power ^  the  method  already  explained,  will 
give  only  the  entire  part  of  the  root,  or  the  root  to  within  less 
than  1.  As  to  the  part  which  is  to  be  added,  in  order  to  com 
plete  the  root,  it  cannot  be  obtained  exactly,  but  we  can  approx- 
imate  to  it  as  near  as  we  please. 

Let  it  be  required  to  extract  the  n*^  root  of  a  whole  number, 

denoted  by  a,  to  within  less  than  a  fraction  —  ;  that  is,  so  near, 

f*mt  the  error  shall  be  less  than   — . 

P 
We  observe,  that  we  can  write 

ap^ 

~~   p'^  ' 

If  we  denote  by  r  the  root  of  the  greatest  perfect  n'*  power  in 

ct  X  P^  r^ 

ap^,  the  number ~  =  a,  will  be  comprehended  between  —  and 

(r+iy  J- 

~ -^  ;   therefore,   the    l/a   will    be    comprised    between   the 

If             tjf  J—  \ 
two  numbers   —   and  ;    and   consequently,  their  difference 

1        .  r 

—   will  be  greater  than  the  difference  between   —  and  the   true 
P  P 

r 
root.     Hence,   —   is   the   required   root   to   within   less  than   the 

fraction  — :   hence, 
P 

To  extract  the  n*^  root  of  a  whole  number  to  within  less  than 
a  fraction  — ,  multiply  the  number  by  p^  ;  extract  the  n*^  root  of 
the  product  to  within  less  than  1,  and  divide  the  result  by  p. 

Extraction   of  the     n*^  Root  of  Fractions, 

145.  Since  the  n^^  power  of  a  fraction  is  formed  by  raismg 
both  terms  of  the  fraction  to  the  n*^  power,  we  can  evidently 
find  the  n^^  root  of  a  fraction  by  extracting  the  n^^  root  of 
both  terms. 


206  ELEMENTS  OF  ALGEBRA.  [CHAP.  VII. 

If  both  terms  are  not  perfect  n*^  powers,  the  exact  n*^  rooi 
cannot  be  found,  but  we  may  find  its  approximate  root  ta 
within   less   than    the  fractional  unit,    as   follows: — 

r  ^ 

l.et     y     represent    the   given  fraction.     If  we  multiply   both 

terms   by 

^"■"■^,       it    becomes,       ■ —  zir  ■ . 

b  b^ 

Let  r  denote  the  n'*  root  of  the  greatest  n*^  power  in  al/^'-^ 
then     — — —     will  be  comprised  between    —    and    ^^—^ — —  ; 

r  a 

and  consequently,    --   will  be  the  n^^  root  of   —    to  within  les» 
o  b 

than  the  fraction     -— - ;     therefore, 

Multiply  the  numerator  by  the  {n—\y^  power  of  the  denomi 
nator  and  extract  the  n^^  root  of  the  product:  Divide  this  root 
by  the  denominator  of  the  given  fraction,  and  the  quotient  will 
he   the   approximate  root. 

When  a  greater  degree  of  exactness  is  required  than  that 
indicated   by    -— ,    extract   the    n*^    root  of  ab"^-^    to  withir   /Jiy 

1  ?•'  r' 

fraction    — ;     and  desio;nate  this  root  by    — .      Now,  since     — 
p  '  '    °  ^      p  '  p 

is  the  root  of  the  numerator  to  within  less  than     — ,    it  fol  ?ws, 

p 

r'     .  1 

that    r;—    is  the  true  root  of  the  fraction  to  within  less  thar.    -— 
op  up 

^  EXAMPLES. 

1.     Suppose  it  were  required  to  extract   the  cube  root  o/   15 

to  within  less  than    -— .     We  have 

15  X  123  =  15  X  1728  =  25920. 
Now,  the   cube   root  of  25920,  to  within   less  thai:   I    is    i^ 
hence,  the  required  root  is, 

12~    12" 


CHAP.   VII.]  EXTRACTION   OF   ROOTS.  207 

2.  Extract  the  cube  root  of  47,  to  within  less  than     --. 

We  have, 

47  X  203  =  47  X  8000  =  376000. 

Now,  the  cube   root  of  376000,  to  within   less  than  1,  is  72  ; 

72         12  1 

hence,        L^-=^=3-^,     to  within  less  than     — . 

3.  Find  the  value  of   \/2b,  to  within   less   than  .001. 

To  do  this,  multiply  25  by  the  cube  of  1000,  or  1000000000. 
which  gives  25000000000.  Now,  the  cube  root  of  this  number. 
is  2920;  hence, 

y^  =  2.920   to  within   less   than   .001. 
Hence,    to    extract    the    cube    root    of   a    whole    number    to 
within  less  than  a  given  decimal  fraction,  we  have  the  following 

RULE. 

Annex  three  times  as  many  ciphers  to  the  number^  as  there  are 
decimal  places  in  the  required  root ;  extract  the  cube  root  of  the 
number  thus  formed  to  within  less  than  1,  and  point  off  from 
the  right  of  this  root  the  required  number  of  decimal  places, 

146t  We  will  now  explain  the  method  of  extracting  the  cube 
root  of  a  decimal  fraction. 

Suppose  it  is  required   to   extract   the   cube   root  of  3.1415. 

Since  the  denominator,  10000,  of  this  fraction,  is  not  a  per 
feet  cube,  make  it  one,  by  multiplying  it  by  100 ;  this  is  equiva 
lent  to  annexing  two  ciphers  to  the*  proposed  decimal^  which  then 
becomes,  3.141500.  Extract  the  cube  root  of  3141500,  that  is, 
of  the  number  considered  independent  of  the  decimal  point  to 
within  less  than  1  ;  this  gives  146.  Then  dividing  by  100,  o> 
^1000000,  and  we  find, 

3^3.1415  =  1.46    to  within  less  than    0.01. 

Hence,  to  extract  the  cube  root  of  a  decimal  fraction,  we  haTi 
the  following 


208  ELEMENTS   OF  ALGEBRA.  [CHAP.   VIL 

RULE. 

Afinex  ciphers  till  the  whole  number  of  decimal  places  is  equal 
to  three  times  the  number  of  required  decimal  places  in  the  root. 
Then  extract  the  root  as  in  whole  numbers,  and  point  off  the  re- 
quired number  of  decimal  places. 

To  extract  the  cube  root  of  a  vulgar  fraction  to  within  less 
than   a  given  decimal  fraction,  the   most   simple   method  is, 

To  reduce  the  proposed  fraction  to  a  decimal  fraction,  continuing 
the  division  until  the  number  of  decimal  places  is  equal  to  three 
times    the  number  required  in  the  root. 

The  question  is  then  reduced  to  extracting  the  cube  root  of 
a  decimal  fraction. 

Suppose  it  is  required  to  find  the  sixth  root  of  23,  to 
within   less   than   0.01. 

Applying  the  rule  of  Art.  144  to  this  example,  we  multiply 
23  by  (100)^,  or  annex  twelve  ciphers  to  23;  then  extract  the 
sixth  root  of  the  number  thus  formed  to  within  less  than  1, 
an(?  divide  this  root  by  100,  or  point  off  two  decimal  places 
on    the   right :   we  thus  find, 

6/23  =  1.68,    to  within  less  than    0.01. 


EXAMPLES. 

1.  Find  the   ^^473   to  within  less  than  ^V-  ^^s,  7| 

2.  Find  the   ^/79   to  within  less  than  .0001.  Ans.  4.2908, 

3.  Find  the   ^ /Ta   to  within  less  than  .01.  Ans,  1.53, 


3 


4.  Find  the   y 3.0041 5   to  within  less  than  .0001. 

Ans,  1.4429. 

5.  Find  the   yO.OOlOl    to  within   less   than   .01. 

Ans,   0.10. 

6.  Find  the   \/\f  to  within  less  than  .001.  Ans,  0.824. 


CHAP.   VIL]  EXTRACTION  OF  ROOTS.  209 

Extraction  of  Roots  of  Algebraic  Quantities, 

147t  Let  us  first  consider  the  case  of  monomials,  and  in  order 
to  deduce  a  rule  for  extracting  the  w*^  root,  let  us  examine  tlie 
law  for  the  formation  of  the  n*^  power. 

From  the  definition  of  a  power,  it  follows  that  each  factor 
of  the  root  will  enter  the  power,  as  many  times  as  there  are 
units  in  the  exponent  of  the  power.  That  is,  to  form  the  w*'' 
power  of  a  monomial. 

We  form  the  n*^  jpower  of  the  co-efficient  for  a  new  co-efficient^ 
and  write  after  this,  each  letter  affected  with  an  exponent  equal  to 
u  times  its  primitive  exponent. 

Conversely,  we  have  for  the  extraction  of  the  w'*  root  of  a 
monomial,  the  following 

RULE. 

Extract  the  n**  root  of  the  numerical '  co-efficient  for  a  new  co- 
efficient, and  after  this   write  each  letter  affected  with  an   exponent 

equal  to  — th  of  its  exponent   in  the  given  monomial;    the  result 
will  he  the  required  root. 


Thus,  yMa^hh^  =  4a35c2 ;      and      \/JWh^  =  ^a'^Pc, 

From  this  rule  we  perceive,  that  in  order  that  a  monomial 
may  be  a  perfect  w**  power: 

1st.  Its  co-eflicient  must  be  a  perfect  n*^  power;   and 

2d.  *  The    exponent   of  each    letter    must   be    divisible  by  w. 

It  will  be  shown,  hereafter,  how  the  expression  for  the  root 
of  a  quantity,  which  is  not  a  perfect  power,  is  reduced  to  its 
simplest  form. 

148.  Hitherto,  in  finding  the  power  of  a  monomial,  we  have 
paid  no  attention  to  the  sign  with  which  the  monomial  may  be 
affected.  It  has  already  been  shown,  that  whatever  be  the  sign 
of  a  monomial,  its  square  is  always  positive. 

14 


210  ELEMENTS  OF  ALGEBRA  LCHAP.   Yll, 

Let  n  be  any  whole  number;  then^  every  pewter  of  an  even 
degree,  as  27^,  can  be  considered  as  the  n*^  power  of  the  square; 
that  is,   {cfiY  =  a?^  :  hence,  it  follows, 

That  every  power  of  an  even  degree^  will  he  essentially  posi 
ttve^  whether  the  quantity  itself  be  positive  .^r  negative. 

This,  {±2a%^cY  z=  +  Ua%^\K 

Again,  as  every  power  of  an  uneven  degree,  2n  +  I,  is  but 
the  product  of  the  power  of  an  even  degree,  2w,  by  the  first 
power ;  it  follows  that, 

Every  power  of  a  monomial^  of  an  uneven  degree^  has  the  same 
sign  as  the  monomial  itself 

Hence,     (+  4.a^h)'^  —  +  64^6^,3  j     and     (  —  4.a%)'^  ■=  —  6^a%K 

From  the  preceding   reasoning,  we   conclude, 

1st.  That  when  the  index  of  the  root  of  a  monomial  is  uneven^ 
the  root  will   be  affected  with  the  same  sign  as   the  monomial. 

Thus, 

1/  +  8a3  =  +  2a ;     Ij  -  Sa^  =  •^2a',     ^  -  ^2a^%^  =  -  2aVK 

2d.    When  the  index   of  the  root  is  even,    and  the   monomial   a 
positive  quantity,   the  root  has  both   the  signs   +  and  — . 
Thus,         *^81a4Z»i2  ^  ^  ^^p .        6^  54^18  _  ^  2a^ 

3d.  Whe7t  the  index  of  the  root  is  even,  and  the  monomial  v.ega» 
live,   the  root  is  impossible; 

For,  there   is  no  quantity  *which,  being  raised  to  a  power  of 
an  even  degree,  will  give   a  negative   result.     Therefore, 
4  /  _^  6  / ^  8  /_  ^ 

lire   symbols   of   operations    which  it  is    impossible   to   execute 
They   are  imaginary  expressions, 

EXAMPLES. 

1.  What  is  the  cube  root  of  Sa^^^c^^?  j^^g^   2a%e^. 

2.  What  is  the  4th  root  of  Sla^^c^^l  Ans,   Sab^c*. 

3.  What  is  the  5th  root  of  —  S2a^c^^d^^  1  Ans.    -^  2ac^d^, 

4.  What  is  the  cube  root  of  —  l^^a^^c^l  Ans,    —  baWe, 


CHAP.    VII.]  EXTRACTION   OF  ROOTS.  211 

Extraction  of  the    n^^  Root  of  Polynomials, 

148t  Let  N  denote  any  polynomial  whatever,  arranged  witb 
reference  to  a  certain  letter.  Now,  the  n*^  power  of  a  poly- 
nomial is  the  continued  product  arising  from  taking  the  poly- 
nomial n  times  as  a  factor:  hence,  the  first  term  of  the  pro- 
duct, wht^n  arranged  with  reference  to  a  certain  letter,  is  the 
n*^  power  of  the  first  term  of  the  polynomial,  arranged  with 
reference  to  the   same   letter. 

Therefore,  the  n*^  root  of  the  first  term  of  such  a  product;, 
will  be  the  first  term  of  the  n*^  root  of  the   product. 

Let  us  denote  the  first  term  of  the  n^^  root  of  N  by  r, 
and  the  following  terms,  arranged  with  reference  to  the  lead- 
ing letter  of  the  polynomial,  by  r',  r'\  /",  &:c.  We  shall 
have, 

N=.  (r  +  r'  +  r"  +  .  .  &c.)»  ; 

or,    if   we   designate  the   sum   of  all   the   terms   after   the  first 

N'=z  (r  -f  sY  =  r^  +nr^~^s  +  &c., 

=  r"  +  nr^-^ir^  +  r"  +  &;c.  )  +  (fee. 

Jf  now,  we  subtract  r^  from  iV^,  and  designate  the  remainder 
by  i?,   we   shall   hav«, 

B  =  N  —  r^  =  nr^-'^r'  +  wr»- V  +  &c., 

which  remainder  will  evidently  be  arranged  with  reference  to 
the  leading  letter  of  the  polynomial;  therefore,  the  first  term 
will  contain  a  higher  power  of  that  letter  than  either  of  the 
succeeding  terms,  and  cannot  be  reduced  with  any  of  them. 
Hence,  if  we  divide  the  first  term  of  the  first  remainder,  by 
n  times  the  (n  —  1)'*  power  of  the  first  term  of  the  root,  the 
quotient  will   be  the  second  term  of  the  root. 

If  now,  we  place  r  +  r'  =  w,  and  denote  the  sum  of  the  suo- 
fieeding  terms  of  the  root  by  5',  we  shall  have, 

iV^=  (u  +  s'Y  =  w"  +  nW'-W  -f  &c 


212  ELEMENTS   OF   ALGEBRA    •  [CHAP.    VLL 

If  now,  we  subtract  w«  from  iV,  and  den  >te  the  remainder  by 
R',  we  shall  have, 

i2'  =  iV^—  M*  r=  w(r  +  ?•')«- V  +  &c., 

=  wr«-i(r"  +  r'"  +  &c.  )  +  &c., 
=  wr'^-V"  +  <^c. 

ll  we  divide  the  first  term  of  this  remainder  by  n  times 
the  {n  —  1)*^  power  of  the  first  term  of  the  root,  we  shall 
have  the  third  term  of  the  root.  If  we  continue  the  operation, 
we  shttll  find  that  the  first  term  of  any  new  remainder,  divided 
by  n  times  the  (n  —  1)**  power  of  the  first  term  of  the  root, 
will  give  a  new  term  of  the  root. 

It  mety  be  remarked,  that  since  the  first  term  of  the  first 
remainder  is  the  same  as  the  second  term  of  the  given  poly- 
nomial,  we  Ci;n  find  the  second  term  of  the  root,  by  dividing 
the  seijond  term  of  the  given  polynomial  by  n  times  the 
(ji  —  1)'*  power  of  the  first  term. 

ITenoe,  for  the  extraction  of  the  n*^  root  of  a  polynomial, 
we  have  the  following 

RULE. 

I.  Arrange  the  given  polynomial  with  reference  to  one  of  its  letters, 
and  extract  the  n*^  root  of  the  first  term;  this  will  be  the  first 
term  of  the  root. 

II.  Divide  the  second  term  by  n  times  the  (n  —  1)'*  power  of  the 
first  term  of  the  root ;  the  quotient  will  be  the  second  term,  of  the  root 

III.  Subtract  the  n*^  power  of  the  sum  of  the  two  terms  already 
found  from  the  given  polynomial^  and  divide  the  first  term  of 
the  remainder  by  n  times  the  {n  —  1)'*  power  of  the  first  term  of 
the   root ;    the   quotient   will   be  the    third   term   of  the   root. 

IV.  Continue  this  operation  till  a  remainder  is  found  equal  to 
0,  OTy  till  one  is  found  whose  first  term  is  not  divisible  by  n  times 
the  (ji  —  1)*^  power  of  the  first  term  of  the  root:  in  the  former  case 
the  root  is  exact,  and  the  given  polynomial  a  perfect  n^^  power  ; 
in  the  latter  case^  the  polynomial  is  an  imperfect  *i*^*  power. 


CHAP.   VII.]  EXTRACTION  OF  ROOTS.  213 

149.  Let   us   apply  the  foregoing  rule  to  the  following 

EXAMPLES. 

1.  Extract  the  cube  root  of  x^-'6x^-i'l5x^--20x^±16x^-'ijx-{^l. 

x^—(yx^+15x^-20x^+l5x^—Qx+l  \x^-2x+\ 
{x^-2xy=x^—6x^-\-l2x^—  Sx^  Sx* 

1st  rem.  3a:*— 12a;3+    &;c. 

{x^-^2x+lY=x^—6x^+l5x^—20x^+15x^—6x+l. 

In  this  example,  we  first  extract  the  cube  root  of  -r^,  which 
gives  a;^,  for  the  first  term  of  the  root.  Squaring  x^,  and  mul- 
tiplying by  3,  we  obtain  the  divisor  3^* :  this  is  contained  in 
the  second  term  —  6x^,  —2x  times.  Then  cubing  the  part  of 
the  root  found,  and  subtracting,  we  find  that  the  first  term  of 
the  remainder  3a;*,  contains  the  divisor  once.  Cubing  the  whole 
root  found,  we  find  the  cube  equal  to  the  given  polynomial. 
Hence,    x^-^2x+l^   is  the   exact  cube  root. 

2.  Find   the   cube   root  of 

x^  +  Qx^  —  40a;3  +  96a:  —  64. 

3.  Find   the   cube   root   of 

Sx^  —  12a;5  +  30a:*  —  25a;3  +  30a;2  —  12a;  +  8. 

4.  Find  the  4th  root  of  16a*  -  96a3a;  +  2lQa?x'^  —  216ax^  -f  81a:* 

16a*-96a3a:+216a2a:2-216aa:3+81a:4     2a-3a: 

(2a— 3a:)*=  1 6a* - 9 6a3a: + 21 6a2a:2- 21 6aa:3  + 8 la:*  4x(2a)3=32a^. 

We  first  extract  the  4th  root  of  16a*,  which  is  2a.  We  then 
raise  2a  to  the  third  power,  and  multiply  by  4,  the  index  of  the 
root ;  this  gives  the  divisor  32a3.  This  divisor  is  contained  in 
the  second  term  —  96a^a:,  —  3a;  times,  which  is  the  second  term 
of  the  root.  Raising  the  whole  root  found  to  the  4th  power 
we  find   the   power   equal    to   the   given   polynomial. 

5.  What   is   the   4th   root  of  the   polynomial, 

Sla^c*  +  IQb^d^  —  9Qa'^cPd^  —  2l6a^c^d  +  216a^c^^d\ 

6.  Find   the  5th  loot  of 

32a;5  —  80a:*  +  SOa:»  «  40a:2  +  10a;  -  1. 


214  ELEMENTS   OF   ALGEBRAr  [CHAP.   VIL 

Transformation  of  Radicals  of  any  Degree, 

150t  The  principles  demonstrated  In  Art.  104,  are  general 
For,  let  "l/a  and  tL/6J  be  any  two  radicals  of  the  ^**  degree, 
iird    denote   their   product   bj  p.     We    shall   have, 

\/^xy^=P    -    .    .    (1). 

By  raising  both  members  of  this  equation  to  the  n'*  power, 
we  find 

(l/«)"  X  ("i/^)"  =i?",       or      ab=ip^\ 

whence,  by  extracting  the  n*^  root  of  both  members, 

y^  =  2)     -    -     -     (2). 

Since  the  second  members  of  equations  (1)  and  (2)  are  the 
same^  their  first  members  are  equal,  whence, 

V^  X  ^fh  —  ^J~ab  :     hence, 
1st.   The  product  of  the  w^*  roots  of  two  quantities^   is  equal  to 
the  n**  root  of  the  product  of  the  quantities. 

Denote  the  quotient  of  the  given  radicals  by  g,  we  shall  have 
^=q     ....     (1); 
and  by  raising  both  members  to  the  n'*  power, 

whence,  by    extracting    the   n'*    root   of   the   two    members,   we 
have,    . 

7?=. (2)- 

Tlie  second  members  of  equations  (1)  and  (2)  being  the  same, 
their   first  members   are   equal,   giving 


'a        »      a 

-  =  y  — ;      hence, 

2cl.   The  quotient  of  the  n*^  roots  of  two  quantities^  is  equal  to 
tJu  n'*  root    of  the  quotient  of  the  quantities. 


CHAP.   VII.1         TRANSFORMATION  OF  RADICALS.  215 

161.  Let  US   apply   the   first  principle    of  article    150,  to   the 
Fimplification  of  the  radicals  in  the  following 

EXAMPLES. 


1.  Take   the.  radical  ^/54a*Pc2.     This  may  be  written, 

2.  In  like  manner, 

3/8a2  =  2y^;      and      t/^a^h^  =  2ah'^c  %/^^^  \ 

3.  Also, 

In  the   expressions,     3a5^2ac2,      2-!/a2,      2a52c \J Stic^, 

each   quantity   placed  before    the   radical,   is   called   a  co- efficient 
of  the  radical. 

Since  we  may  simplify  any  radical  in  a  similar  manner,  we 
have,  for  the  simplification  of  a  radical  of  the  n^^  degree,  the 
following 

RULE. 

Resolve  the  quantity  under  the  radical  sign  into  two  factors^  one 
of  which  shall  be  the  greatest  ^perfect  n*^  power  which  enters  it; 
extract  the  n^^  root  of  this  factor^  and  write  the  root  without  the 
radical  sign,  under  which,  leave  the  other  factor. 

Conversely,  a  co-efficient  may  he  introduced  under  the  radical 
sipn,  by  simply  raising  it  to  the  n^^  power,  and  writing  it  as  a 
factor  under  the  radical  sign. 

Thus,     Zab  \f^M^  =  3/27^3^3  x  ^2^  -  y  h^a^b'^c^. 

152.  By  the  aid  of  the  principles  demonstrated  in  article  143, 
we   are   enabled   to   mafke   another   kind  of  simplification. 

Take,  for  example,  the  radical  %/~^',  from  the  principles  re- 
ferred  to,  we  ha^e. 


^■v^ 


£/4a2  —  \/^/4a^ 


216  ELEMENTS  OF  ALGEBRJf.  [CHAP.   VII. 

oad  as  the  quantity  under  the  radical  sign  of  the  second  degree 
is  a  perfect  square,  its  root  can  be  extracted :   hence, 

Li  like  manner, 
In  general, 


that  is,  when  the  index  of  a  radical  is  a  multiple  of  any  n*iml>er 
n,  and  the  quantity  under  the  radical  sign  is  an  exact  n^^  power, 
We  can^  without  changing^  the  value  of  the  radical^  divide  its  index 
by  n,  and  extract  the  n*^  root  of  the  quantity  under  the  sign, 

153.  Conversely,  The  index  of  a  radical  may  he  multiplied  by 
any  number^  provided  we  raise  the  quantity  under  the  sign  to  a 
power  of  which  this  number  is  the  exponent. 

For,  since  a  is  the  same  thing  as  !L/^,  we  have, 


V^=  yV^^ 


154«  The  last  principles  enable  us  to  reduce  two  or  more 
radicals  of  different  degrees,  to  equivalent  radicals  having  a  com- 
mon index. 

For  example,  let  it  be  required  to  reduce  the  two  radicals 

y^    and    yT^T^) 

to    the   same   index. 

By  multiplying  the  index  of  the  first  by  4,  the  index  of  the 
second,  and  raising  the  quantity  2a  to  the  fourth  power;  then 
multiplying  the  index  of  the  second  by  3,  the  index  of  the 
first,  and  cubing  a  4-  6,  the  value  of  neither  radical  vill  bp 
changed,  and  the  expressions  will  become 

^2^  =  iy^2%^  =  ly  Fe^;     and     \/ {a  +  b)  =  ^^W+W^ 
and  similarly  for  other  radicals:   hence,  to   reduce   radicals   to  a 
common  index,  we  have  the  following 


CHAP.    VII.]         TKANSFORMATION"   OF  RADICALS.  217 

RULE. 

Multiply  the  index  of  each  radical  hy  the  product  of  the  indices 
of  all  the  other  radicals^  and  raise  the  quantity  under  each  radical 
sign  to  a  power  denoted  by  this  product. 

This  rule,  which  is  analogous  to  that  givec  for  the  reduction 
of  fractions  to  a  common  denominator,  is  susceptible  of  similar 
modifications. 

For  example,  reduce  the  radicals  ^ 

to  a  common  index. 

Since  24  is  the  least  common  multiple  of  the  indices,  4,  6,  and 
8,  it  is  only  necessary  to  multiply  the  first  by  6,  the  second  by 
4,  and  the  third  by  3,  and  to  raise  the  quantities  under  each  rad 
ical  sign  to  the  6th,  4th,  and  3d  powers,  respectively,  which  gives 

y^=  2^/06";     %fSb  z=z'^\f^h^,     y  a^  -Y  h"'  =.'^\J {a'^  IP-f. 

Addition  and  Subtraction  of  Radicals  of  any  Degree, 

155.  We  first  reduce  the  radicals  to  their  simplest  form  by 
the  aid  of  the  preceding  rules,  and  then  if  they  are  similar^  in 
order  to  add  them  together,  we  add  their  co-efficients^  and  after 
this  sum  write  the  common  radical;  if  they  are  not  similar,  the 
addition   can   only   be   indicated. 

Thus,  Z%fb  +  2^^=^%/b. 

EXAMPLES. 


1.  Find  the  sum  of  ^48^   and    b^lba.       Ans.  9b ^a, 

2.  Find  the  sum  of   Z\f^    and    23/2a".         Ans,  S^/Sa. 

3.  Find  the  sum  of   2  y^    and    3  J~S,  Ans.  9  y^. 

155*.  In   order  to   subtract    one    radical  from    another  whou 
they  are   similar, 

Subtract  the  co-efficient  of  the  subtrahend  from  the  co-efficient  if 
Hie  minuend^  and  write  tlis  difference  before  the  common  radical^ 


218  ELEMENTS   OF  ALGEBRA  .CHAP.    VIL 

Thus,  3a  %/h  -  2c  \fb  =  (3a  -  2c)  yT; 

but,  2ah  yTcd^  5a6  ,J~c    are  irreducible. 

1.  From     J^8a36+  16a^     subtract     y'  6*  +  2a63. 

.4n5.  (2a  -  Z>)  y6  +  2a. 

2.  From     3  ^"402"    subtract    2^/2a.  ^ns.      3/2a. 

Multiplication  of  Radicals  of  any  Degree. 

156.  We  have  shown  that  all  radicals  may  be  reduced  to 
equivalent  ones  having  a  common  index;  we  therefore  suppose 
this   transformation   made. 

Now,  let  a  "tTb  and  c '^Td  denote  any  two  radicals  of  the 
same  degree.     Their  product  may  be  denoted  thus, 

a  !^  X  c  yT; 
or  since  the  order  of  the  factors  may  be  changed  without  affect- 
ing the  value  of  the  product,  we  may  write  it, 

ac  X  y^X  \/~d    or  (Art.  150),  since     ^Tx  y^=  \f^\ 
we  have  finally, 

al/Tx  c'lTd—  aclfbd\ 
hence,  for  the  multiplication  of  radicals  of  any  degree,  we  have 
the  following 

RULE. 

I.  Meduce  the  radicals  to  equivalent  ones  having  a  common  index, 

IT  Multiply  the  co-efficients  together  for  a  new  co-efficient ;  after 
this  write  the  radical  sign  with  the  common  index,  placing  under 
it  the  product  of  the  quantities  under  the  radical  signs  in  the  two 
factors;  the  result  is    the  product  required. 


1.  The  product 

EXAMPLES. 

3     /«2  _|_  12 

W       c        X 

3^  [  /(«'  +  *')'  _ 

— V'""i"'' 

6w'  (g^  +  P) 


CHAP.   VII.j  TRANSFOKMATION   OF  RADICALS.  219 

2.  The  product 

3a ^/^8^X  26 y^4^  =  Qab ^32^  =  \2a?b ^^/2c. 

3.  The  product 

4  V  3"  ^  TV   7  ""  16V2r 

4.  The  product 

3ayTx  55y^=  15c-^  X  ''\/~W^. 

5.  Multiply   y2x^     by     y^x^/^. 

6.  Multiply     2^15     by     3  3^/TO. 

^7i5.  6  5/337500. 

^  /2~  /3" 

7.  Multiply    4W—    by    2\/-~-. 

^"^-  ^  V  256- 

8.  Multiply   .^/^     y^     and     y^,     together. 

^/i5.    1^648000. 

»     /T       ^     /T  y— 

9.  Multiply     w-TT)    \/ir    ^^^     ^V  ^'     together. 

\      O  \      tit  ^ 

10.  Multiply    (4y^+5^)     by     (yi+2,/1). 

43   ,   13    rz: 

Division   of  Radicals   of  any  Degree. 

157i  We  will  suppose,  as  in  the  last  article,  that  the  radicala 
have  been  reduced  to  equivalent  ones  having  a  common  inlex. 

Let  cilfb^  and  Clfd  represent  any  two  radicals  of  the 
r,  *  degree.  The  quotient  of  the  first  by  the  second  may  be 
written, 

o\/~d    r  v/s' 


220  ELEMENTS   OF  ALGEBRA.  [CHAP.   VIL 

v^   "  rv 

or,  since  ■^  =  \/-r  (Art.  150),   re  have, 

n^        V    G?     ^ 

c  "^fd       ^      V  c? 
Hence,    to   divide   one   radical   by   another,  we  have   the   fo 
lowing 

RULE. 

I.  Reduce  the  radicals  to  equivalent  ones  having  a  common  indea^, 

II.  Divide  the  co-efficient  of  the  dividend  hy  that  of  the  divi- 
sor for  a  new  co-efficient;  after  this  write  the  radical  sign  with 
the  common  index,  and  place  under  it  the  quotient  obtained  hy 
dividing  the  quantity  under  the  radical  sign  in  the  dividend  hy  that 
in  the  divisor ;    the  result  will  he  the  quotient  reqwred, 

EXAMPLES. 


/ — : ^   /a' 

1.  What  is  the  quotient  of  c  yaW  +  h^  divided  by  c?  w  - 

^       IJaW  +  6^  __   c  3    /86  (^2^2  ^  ^,4)  ^  2c5  3   la^  +  h'^ 
d   ^3    r^^rT^~'d\l       a2_^2       -"TV  a2-62' 

V-sT- 

2.  Divide    2^x\/4    by     i\/^X?/^. 


,2_62^ 


3.  Divide   y^  X2'/S     by   y^4  ^^  x  y^ 


4.  Divide    l>/i    by     (^2  + 3^). 

5.  Divide  1  by    \/^+\/J, 


Afis, 

4  ly  288. 

v^ 

Ans. 

1  12    /2' 

2  V   3' 

^...  ~ 

CRAP     Vll.I         TRANSFORMATION  OF  RADICALS  221 

6.  Divide     \f^+  \fh    by     ^/11  -  1^ 


a  —  6 

Formation  of  Powers  of  Badicals  of  any  Degree.  i 

158»  Let  a  l/^  represent  any  radical  of  the  n^^  degree. 
Then  we  may  raise  this  radical  to  the  m'^  power,  by  taking 
It  m   times   as   a   factor;  thus, 

(X'%/hXa'l^ a  ^^/^ 

But,  by  the  rule  for  multiplication,  this  continued  product  is 
equal   to   a^  \fi^)  whence, 

(a  "l^Y  =  ^"^  V^         .    -    -    .     (1). 

We  have  then,  to  raise  a  radical  to  any  power,  the  following 

RULE. 

Raise  the  co-efficient  to  the  required  power  for  a  new  co-efficient ; 
after  this  write  the  radical  sign  with  its  primitive  index,  placing 
under  it  the  required  power  of  the  quantity  under  the  radical 
sign  in  the  given  expression ;    the  result  will  be  the  power  required, 

EXAMPLES. 

2.     (3  3^/2^)5  ziz  35   ^([2^)5  =  243  ^/32i^=  486a  ^/"i^. 

When  the  index  of  the  radical  is  a  multiple  of  the  expo- 
nent of  the  power  to  which  it  is  to  be  raised,  the  result  can 
be   simplified. 

For,    1^/^=  WySo"  (Art.  152):    hence,  in   order   to  square 

i/2a,   we  have   only  to  omit  the  first  radical  sign,  which  gives 

Again,  to  square     ^/^,    we  have    ^/36  =  \/l/^-    hence, 
{%fSby=:l/U\     hetce, 


222  ELEMENTS   OF   ALGEBHA.  [CHAP.   VII. 

When  the  index  of  the  radical  is  divisible  by  the  exponent  of 
the  power  to  which  it  is  to  be  raised,  perform  the  division,  leaving 
the   quantity  under   the   radical  sign   unchanged. 

Extraction  of  Roots  of  Radicals  of  any  Degree. 

159.  By  extracting   the   m*^   root   of  both   members  of  equo 
tion  (1),  of  the   preceding   article,  we  find, 


Whence  we   see,  that  to  extract  any  root  of  a  radical  of  any 
degree,  we   have   the  following 


RULE. 


Extract  the  required  root  of  the  co-efficient  for  a  new  co-efficient ; 

after    this   tvrite    the   radical  sign  with   its  primitive   index,   under 

which  place   the  required   root   of  the   quantity   under    the   radical 

sign  in    the  given  expression;    the  result  will  be  the  root  required. 


EXAMPLES. 


1.  Find   the  cube  root  of  81^/27.  Ans,   2l/3". 

2.  Find   the   fourth   root  of  —3/256.  Ans,   ~  i/i. 

lo^  2   V 

159^.  If,  however,  the  required  root  of  the  quantity  under  the 
radical  sign  cannot  be  exactly  found,  we  may  proceed  in  the 
following  manner.  If  it  be  required  to  fmd  the  m'*  root  of 
cv/^  the  operation  may  be  indicated  thus, 


but    \J  'u~d  —  "^'iTd^    whence,    by    substituting   in    the   previous 

equation, 

\/c  \fl  =  '^^"'y^  : 

Consequently,  when  we  cannot  extract  the  required  root  of  the 
quantity  under  the  radical  sign. 


CHAP.   VII.]         TRANSFORMATION    OF   RADICALS.  223 

Extract  the  required  r^ot  of  the  co-efficient  for  i  new  co-efficient; 
after  this,  write  the  radical  sign,  with  an  index  equal  to  the  pro- 
duct  of  its  jprimitive  index  by  the  index  of  the  required  root, 
leaving  the  quantity  under  the  radical  sign  unchanged, 

e:^amples. 

1.       yV^^^^V^;    and,   Yw^^V^'- 

When  the   quantity   under  the   radical  is  a  perfect  power,  of 

tlie  degree  of  either  of  the  roots  to  he  extracted^  the  result  can  be 

simplified. 


Thus,  y^y^e^^y^i 


In  like  manner,     \/  y^^  _  W  y^^  —  y^. 

2.  Find  the  cube  root  of  ^-y/^-  ^^^-   T  V^* 

3.  Fmd  the  cube  root  of  --y/2^.  Ans,   —  y^2^. 


Different  Roots  of  the  same  Power, 

160t  The  rules  just  demonstrated  depend  upon  the  principle^ 
that  if  two  quantities  are  equal,  the  like  roots  of  those  quantities 
are  also  equal. 

This  principle  is  true  so  long  as  we  regard  the  term  root 
in  its  general  sense,  but  when  the  term  is  used  in  a  restricted 
sense,  it  requires  some  modification.  This  modification  is  parti- 
cularly necessary  in  operating  upon  imaginary  expressions,  which 
are  not  roots,  strictly  speaking,  but  mere  indications  of  opera- 
tions which  it  is  impossible  to  perform.  Before  pointing  out 
these  modifications,  it  will  be  shown,  that  every  quantity  has 
more  than  one  cube  root,  fourth  root,  &c. 

It  has  already  been  shown,  that  every  pantity  has  two  square 
roots,  equal,  with  contrary  signs. 


224  ELEMENTS   OF  ALGEBRi..  [CHAP.   VII. 

1.  Let  X  denote   the   general    expression  for  the  cube  root  of 

a^  and  let  p  denote  the  numerical  value  of  this   root ;    we  have 

the  equations 

x^  =z  a,     and    x"^  =  j9^ 

The  last   equation  is  satisfied  by  making   a;  =:^. 

Observing  that  the  equation  x^  =  ^^  can  be  put  under  the  form 
jg.3  __  ^3  __  Q^  and  that  the  expression  x^  —  p^  is  divisible  by 
X  '  —  p,  giving  the  quotient,  x"^  +  px  -\-  p^^  the  above  equation  can 
be  placed  under  the  form 

{x  —p)  {x^  -{-px  +^2)  =  0. 

Now,  every  value  of  x  that  will  satisfy  this  equation,  will 
satisfy  the  first  equation.  But  this  equation  can  be  satisfied  by 
supposing 

x  —  p  =  0^    whence,     x  =^; 
or  by  supposing 

X^  +  px  -\-  p'^'  zzz  0^ 

from  which  we  have, 

hence,  we  see,  that  there  are  three    different   algebraic   expressions 
for  the  cube  root  of  a,  viz : 

2.  Again,  solve   the   equation 

x^  =  p\ 

ill  which  p  denotes  the  arithmetical  value  of  i/ci. 
This  equation  can  ba  put  under  the  form 

x^—p^=0', 
which   reduces   to 

{x^^p^){x^+p^)z=0; 
and   this  equation  can  be  satisfied,  by  supposing 
a;2  __  ^2  -_  Q .      whence,     x  =  dt  p; 
or   by   supposing 


z^  -i-  p^  =  0,     whence,     x  =  ±.^  —p'^  =  ±  p^ 


CHAP.  VII.]         TKANSP'ORMATION  OF  RADICALS.  226 

We  therefore  obtain  four  different  algebraic  expressions  for  the 
fourth  root  of  a, 

3.  As  another  example,  solve  the  equation 

ic®  —  ^^  =  0. 
This  equation  can  be  put  under  the  form 

vrhich  may  be  satisfied  by  making   either  of  the  factors  equal 
to  zero. 

But,  a;3  __  |j3  -_  0^     gives 


*a;=^,     and    x  =  jpy -^ j. 

And  if  in  the   ei^uation    a;3+j!53  — o,   we  make   ^  =  —  j?',   it 
becomes  x^  —p'^  =  0,  from  whicL  we  deduce 


x=zp\     and    x=p'y ^ j  ; 

or,  substituting  for  p'  its  value  --^, 

=-H — 2: — y 


rr  =  —  jE?,     and     x 


Therefore,  x  in  the  equation 

x^  — jp6  -.  0, 

and  consequently,  the  6th  root  of  a,  admits  of  six  different  alg^ 
hraic  expressions.     If  we  make 

a  = ^ ,     and    a'= ^ , 

these  expressions  become 

jp,     op,     a'^,     —- p^— ap^-^  a'p. 

It  may  be  demonstrated,  generally,  that  there  are  as  many 
different  expressions  for  the  n*^  root  of  a  quantity  as  there  are 
units  in  n.  If  n  is  an  even  number,  and  the  quantity  is  posi- 
tive, two  of  the  expressions  will  be  real,  and  equal,  with  con- 
trary elgns ;  all  the  rest  will  be  ini^inary :  if  the  quantity  is 
negative,  they  will  all  be  imaginary. 

15 


228  ELEMENTS  OF  ALGEBHA.       LCHA.P.  VII. 

If  n  is  odd,  one  of  the  expressions  will  be  real,  and   all  the 
rest  will  be  imaginary. 

161.  If  in  the  preceding  article  we  make  a  =  1,  we  shall  find 
the  expressions  for  the  second,  third,  fourth,  &;c.,  roots  of  1. 
Thus,     +  1    and  —  1    are  the  square  roots  of  1. 

Also,      +  1,  ^ ,     and      ^ , 

are  the  cube  roots  of  1 : 


And  +  1,  •  1,  +y^—  1  and  — -/—  1,  are  the  fourth 
roots  of  1,  &;c.,  &;c. 

Bules  for  Imaginary  Expressions, 

162.  We  shall  now  explain  the  modification  of  the  rules  for 
operating  upon  radicals   when   applied  to   imaginary  expressions. 

The  product  of  V  —  a  by  ^  —  a,  by  the  rule  of  Art.  156, 
would  be  ^  +  a^.  Now,  -/~+a^  is  equal  to  ±  a,  whence  there 
is  an  apparent  uncertainty  as  to  the  sign  of  a.  The  true  pro- 
duct, however,  is  —  a,  since,  from  the  definition  of  the  square 
root  of  a  quantity,  we  have  only  to  omit  the  radical  sign,  to 
obtain  the  quantity. 

Again,  let  it  be  required  to  form  the  product 
By  the  rule  of  Art.  156,  we  shall  have 

but   the   true    result   is    — y^«^,  so    long    as   both  the  radicals 
^  —a  and  y^  —  6  are  afiected  with  the  sign   +. 

For,     ,yj~^^a  ==y^. ^  —  1  ;     and     ^  —  h  zzi^Jh",^  —  1  ^ 
hence, 


—  yf^  X  —  1  =  — yaF. 


CHAP.   YII.J        TRANSFORMATION   OF  RADICALS.  227 

In  a  similar  manner,  we  treat  all  other  imaginary  expressions 
of  the  second  degree ;  that  is,  we  first  reduce  them  to  the  form 
of  ay/  —  1,  in  which  the  co-efficient  of  y/  —\  is  real,  and  then 
proceed  as  indicated  in  the  last  article. 

162*.  For  convenience,  in  the  application  cf  the  preceding 
principle,  we  deduce  the  different  powers  of  -y/  —1,  as  follows  r 


The  fifth  power  is  evidently  the  same  as  the  first  power ;  the 
sixth  power  the  same  as  the  second;  the  seventh  the  same  as 
the  third,  and  so  on,  indefinitely. 

163.  If  it  is    required    to    find   the   product   of   1/—  a    and 
1/  —  6,  we  should  get,  by  applying  the  rule  of  Art,  156. 
1/  —  a  X  1/  —  6  =  */  +  ab,  but  this  is  not  the  true  result 
For,  placing  the  quantities  under  the  form 

\Ax\^=n:  and  y^xi/^=n:, 

and  proceeding  to  form  the  product,  we  find 

since,    (^  —  l)^  =  l\/ ^  —  l\    z=^  —  1  from  the  definition  of 
a  root. 

Hence,  generally,  when  we  have  to  apply  the  rules  for  radi- 
cals  to  imaginary  expressions  of  the  fourth  degree,  transform 
theixi,  so  that  the  only  factor  under  the  radical  sign  shall  be 
—  1,  and  then  proceed  as  in  the  above  example. 

Let  us  illustrate   this   remark,  by  showing  rhat  -r-^^- 

is  an  expression  for  the  cube  root  of  1,  or  that,  ir   the  restricced 
feense^  it  is  a  cube  root  of  1. 


228  ELEMENTS   OF  ALGEBRA.  iCHAP.    VTT 

We  have 

8  • 

-1+3  yS".  ^/^^  -3X-3-3V3.  -y/^T        8 
^  8  "■  8  ~ 


—  1  —  ^  __  3 

In  like  manner,  we  may  show,  that    ^ is  another 

expression  for  the  cube  root  of  1,  when  understood  in  the 
restricted  sense.  It  may  be  remarked  that  either  of  these  ex- 
pressions is  equal  to  the  square  of  the  other,  as  may  easily 
be  shown. 

Of  Fractional  and  Negative  Exponents. 

164,  We  have  yet  to  explain  a  system  of  notation  by  means 
of  which  operations  upon  radical  quantities  may  be  greatly 
simplified. 

We  have  seen,  in  order  to  extract  the  n^^  root  of  the  quan- 
tity a*",  that  when  i/C  is  a  multiple  of  w,  we  have  simply  to 
divide  the  exponent  of  the  power,  by  the  index  of  the  root  to 
be   extracted,  thus, 

n     I  m 

When  m  is  not  a  multiple  of  w,  it  has  been  agreed  to 
retain   the  notation. 


these  two  being  regarded  as  equivalent  expressions,  and  botli 
indicating  the  ^'^  root  of  the  m*^  power  of  a,  or  what  is  the 
same  thing,  the  w'^  pc  wer  of  the  n^^  root  of  a ;  and  generally, 

Wheyi  any  quantity  is  written  with  a  fractional  exponent,  the 
numerator  of  the  fraction  denotes  the  power  to  which  the  quantity 
is  to  be  raised,  and  the  denominator  indicates  the  root  of  this 
power  which  is   to   be   extracted. 


CHAP.   VII.]  '        THEORY  OF  EXPONENTS.  229 

165i  We  have  also  seen  that  a*^  maj  be  dinded  by  a", 
when  m  and  n  are  whole  numbers,  by  simply  subtracting  n 
from   m,   giving 

in  which  we  have  designated  the   excess   of  m   over  n   by  p. 

Now,   if   n   exceeds   m^   p    becomes  negative,   and  the   exact 

division   is   impossible  ;    but  it    has  been  agreed   to  retain  the 

notation 

a"* 

a« 
But  when  m  <  w,  in   the  fraction, 

a*" 

^' 

we  may  divide  both  terms  by  a*",  and  we  have 

a*^  _    1       __  1    . 

a"        a^-^       aP  ' 

hence,  a~P  is  equivalent  to   — ,     and  both  denote    the    recipro- 
cal of  aP. 

We  have,  then,  from  these  principles,  the  following  equiva- 
lent  expressions,  viz. : 

i_ 
*i/a  equivalent  to     a". 

m 

*L/a^  or  {"ifaY     "  ^" • 

1 


ar^. 


1  «  fT  -1 

— F=    or  a/—       "  a    ». 

ya  V    a 

-7=  or  W—       "  a    «. 

166.  It  has  been   shown   above  that    —  =  a-^  :     if   now   we 

a* 

divide   1   by   both    members   of   this    equation,   we  shall  ha\e, 
a*  =  -— -  :    hence   we  conclude  that. 


230  ELEMENTS   OF   ALGEBKA.  [CHAP.   VII 

Any  factor  may  he  transferred  from  the  numerator  to  the  de- 
nominator, or  from  the  denominator  to  the  numerator,  hy  changing 
the  sign   of  its   exponent. 

167.  It  may  easily  be  shown  that  the  rules  for  operatiug 
upon  quantities  when  the  exponents  are  positive  whole  numbers, 
are   equally   applicable  when  they  are  fractional  or  negative. 

In  the  first  place,  it  is  plain  that  both  numerator  and 
denominator  of  the  fractional  exponent  may  be  multiplied  by 
the  same  quantity  without  altering  the  value  of  the  expression, 
since  by  definition  the  m*^  power  of  the  m*^  root  of  a  quan- 
tity is  equal  to  the  quantity  itself.  This  principle  enables  us 
to  reduce  quantities,  having  fractional  exponents,  to  equivalent 
ones   having   a   common   denominator. 

Let  it  be  required  to  find  the  product  of  a*  and  a** 

m  r  ms  nr 

We  have,  o^  X   a^  =  a^  X  a"*' 


or  (Art.  164),        «y^a"'«  X  «ya^= '»ya^*  +  «'" ' 

ms  -\-vr 

This  last  result  is  equivalent  to   a    "*     '     hence, 

m  r  77JS  +  wr  ^ 

a^  X  a  *  =  a    «*      ' 

the.  same   result  that   would   have   been   obtained   by  the   appli- 
cation of  the   rule   for   the    multiplication    of   monomials,    when 
the    exponents   are   positive   whole   numbers. 
If  both   exponents   are   negative,  we  shall  have, 

I  _^  _^  11  1  ms-^-nr 

^  m  r  ms  +  -nr 

a^       a*        a    »»^ 
Jf  one   of  the  exponents  is  positive,  and  the  other  negative, 
-ve  shall  have, 

m  r  m  -,  ms  -i 

a*  X  a"*  =z  a«  x  ~  =  a«»  X  -^  , 


CHAP    VII.]  THEORY  OF  EXPONENTS.  231 

whence,        «ya^  X     W  -^  =    V  ~~^  ~  ns^a'^i^nr  =  „  "•    ' 


TO5  — nr 


and  finally,  a»»  X  a    *  =  a  "* 

We  have,  therefore,  for  the  multiplication  of  quantities  when 
the  exponents  are  negative  or  fractional,  the  same  rule  as  when 
they  are  positive  whole  numbers,  and  consequently,  the  same 
rule  for  the  formation  of  powers. 

EXAMPLES. 
3._1  23  lli_i 

1.  a^b    2cr"i  xa?b''c^  =a^  b^c    \ 

2.  Sa-^'^  X  2a"^6  V  =  QaT  ^  6V. 

1 

4.  Find  the  square  of  f  a^ . 

We  have,  (|a^)'  =  (f)^  x  a^"" '  =  |ai 

5.  rind  the  cube  of  ^a  .  -4w5.    ^jO^. 

m  r 

168.  Let  it  be  required  to  divide  a"  by  a».     We  shall  have, 

TO  m 

~  =a«  X  a    *'      or  (Art.  167),      —  =  «    "* 
a^  a* 

If  both  expon^-jts  are  negative. 


=  a   »»  X  a*  =  a    «*    '     by  the  last  art:'"cle. 

a""* 
kf  one  exponent  is  negative, 

TO 

—  TO  ^  ms-\'  rn 

^ —  =  a «  X  "**  =  «    ***    '    by  the  preceding  article. 


282  ELEMENTS  OF  ALGEBRA.  [CHAP.   VH. 

Hence,  we  see  that  the  rule  for  the  division  of  quantities, 
with  fractional  exponents,  is  the  same  as  though  the  exponents 
were  positive  whole  numbers;  and  consequently  we  have  the 
same  rule  for  the  extraction  of  roots,  as  when  the  exponents  are 
positive   whole  numbers. 

EXAMPLES. 

3.  a^x5*-r-a"M=  A"* 

4.  Divide     S2a^b^c^     by     SaH^c"^'.  Ans.  4.a^bc*. 

5.  Divide     Ma^K'^     by     32a~9^^~^c""l  Ans.   2a}^h\ 
6. 


169.  We  see  from  the  preceding  discussion,  that  operations  to 
be  performed  upon  radicals,  require  no  other  rules  than  those 
previously  established  for  quantities  in  which  the  exponents  are 
entire.  These  operations  are,  therefore,  reduced  to  simple  oper 
ations  upon  fractions,  with  which  we  are  already  familiar. 

GENERAL     EXAMPLES. 

1.     Reduce     --^- ^     •     to  its  simplest  terms. 

i-/2 


2.     Reduce      -( j-  >      to  its  simplest  terms. 

(  2^2(3)*   ) 


Ans.    4  j/sT 


^"*-   314  V^- 


CHAP    VII.J  THEORY  OP  EXPONENTS.  233 


3.    Reduce    /    I  izL+jL^  (      to  its  simplest  terms. 


4.     What  is  the  product  of 

a^  ^a^b^  +  Jb^ +  ab  +  ah^+b^,   by  a*  -  6*. 


Ans,    a^  —  b'^. 
5.     Divide     a^-^a^"^  -  a*6  +  b^,   bj   a*  -  T^. 

170.  If  we  have  an  exponent  which  is  a  decimal  fraction,  as, 
for  example,  in   the  expression  10  *  ^^^    from  what  has  gone  bp. 

301  

fore  the  quantity  is  equal  to  (10)^*^^°'  or  to  ioo^(io)3oi^  the 
value  of  which  it  would  be  impossible  to  compute,  by  any  process 
yet  given,  but  which  will  hereafter  be  shown  to  be  nearly  equal 
to  2.  In  like  manner,  if  the  exponent  is  a  radical,  as  VS^  V^TT, 
&c.,  we  may  treat  the  expression  as^  though  the  exponents  were 
fractional^  since  its  values  may  be  determined,  to  any  dogree  of 
exactness,  in  decimal  terms. 


CHAPTER  VIL. 

OF     SERIE.$ AR  THMETICAL     PROGRESSION GEOMETRICAL     PROPORTION     AND 

PROGRESSION RECURRING     SERIES BINOMIAL    FORMULA SUMMATION      OF 

SERIES PILING    SHOT   AND     SHELLS. 

171  •  A  SERIES,  in  algebra,  consists  of  an  infinite  number  of 
terms  following  one  another,  each  of  which  is  derived  from 
one  or  more  of  the  preceding  ones  by  a  fixed  law.  This  law 
is  called  the  law  of  the   series. 

Arithmetical  Progression, 

172.  An  ARITHMETICAL  PROGRESSION  is  a  scrics,  in  which  each 
term  is  derived  from  the  preceding  one  bj  the  addition  of  a 
constant   quantity  called   the   common   difference. 

If  the  common  difference  is  positive^  each  term  will  be  greater 
than  the  preceding  one,  and  the  progression  is  said  to  be  in 
creasing. 

If  the  common  difference  is  negative^  each  term  will  be  less 
than  the  preceding  one,  and  the  progression  is  said  to  be 
decreasing. 

Thus,  ...  1,  3,  5,  7,  .  .  .  &c.,  is  an  increasing  arithmetical 
progression^  in  which   the   common   difference   is   2  ; 

and 19,  16,  13,  10,  7,  ...  is   a    decreasing  arithmetical 

progression^  in  wliich   the   common   difference  is    —  3. 

173.  When  a  certain  number  of  terms  of  an  arithmetical 
progression  are  considered,  the  first  of  these  is  called  the  first 
term  of  the  progression^  the  last  is  called  the  last  term  of  the 
progression,  and  both  together  are  called  the  extremes.  All  the 
terms  between  the  extremes  are  called  arithmetical  means.  An 
arithmetical  progression  is  often  called  a  progression  hg  differences. 


CHAP.   VIII.]  ARITHMETICAL  PROGRESSION.  235 

174»  Let  d  represent  the  common  difference  of  the  Arithmeti- 
cal  progression, 

a.b.c.e.f,g,h,1c^  &c., 
which  is   written  bj  placing  a  period   between  each  two  of  the 
terms. 

From   the   definition  of  a   progression,  it  follows  that, 
h  =za  +  d^     c  =  54-c?  =  a  +  2o?,     e  =  c  +  ef=a  +  3c?; 
Mid,    in    general,    any   term   of  the   series,  is   equal  to    the  first 
term  plus  as  many  times    the   common  difference  as  there  are  pre- 
ceding terms. 

Thus,  let  I  be   any  term,  and  n  the  number  which  marks  the 
place  of  it.     Then,  the   number  of  preceding  terms  will   be  de- 
noted by  ^  —  1,  and  the  expression  for  this  general  term,  will  be 
I  z=i  a  +  (n  —  l)d. 

If  d  is  positive,  the  progression  will  be  increasing ;  hence, 

In  an  increasing  arithmetical  progression,  any  term  is  equal  to 
the  first  term,  plus  the  product  of  the  common  difference  by  the 
number  of  preceding  terms. 

If  we  make  71  =  1,  we  have  ^  =  a ;  that  is,  there  will  be 
but   one   term. 

If  we  make 

w  =  2,     w^e   have     Z  =  a  +  c? ; 
that  is,  there  will  be  two  terms,  and  the  second  term  is  equal 
to  the  first  plus   the  common  difference. 

EXAMPLES. 

1.  If   a  =  3    and    c?  =  2,    what  is  the  3d  term?        Ans.  7. 

2.  If   a  =  5    and    c?  =  4,    what  is  the  6th  term  ?     Ans,  25. 

3.  If   a  =  7    and    d—^,    what  is  the  9th  term  1     Ans,  47. 

The   formula, 

Z  =  a  +  (71  -  1)  c/, 
serves   to   find   any   term   whatever,    without   determining    those 
which   precede  it. 


286  ELEMENTS   OF   ALGEBBA.  [CHAP.   Tin. 

Thus    to  find  the  50th  term  of  the   progressicn, 
1  .  4  .  7  .  10  .  13  .  16  .  19,  .  . 
we  have,  Z  =  1  +  49  X  3  =  148. 

And  for   the    60th   term  of  the   progression, 

1  .  5  .  9.  13  .  17  .  21  .  25,  .  .  . 
we  have,  Z  =  1  +  59  X  4  =  237. 

174*»  If  d  is  negative,  the  progression  is  decreasing,  and  the 
formula  becomes 

Izi^a  —  {n  —  \)d\     that  is. 

Any  term  of  a  decreasing  arithmetical  progression^  is  equal  to 
the  first  term  plus  the  product  of  the  common  difference  by  the 
number   of  preceding  '  terms, 

EXAMPLES. 

1.  The  first  term  of  a  decreasing  progression  is  60,  and  the 
common   difference    —  3 :   what   is   the   20th   term  1 

l^a--{n-l)d    gives     Z  =  60  -  (20  -  1)3  =  60  -  57  =  3. 

2.  The  first  term  is  90,  the  common  difference  —  4 :  what 
IS   the    15th   term?  Ans,  34. 

3.  The  first  term  is  100,  and  the  common  difference  —  2  • 
what  is   the   40th   term  ?  Ans,  22. 

175«  If  we  take   an  arithmetical  progression, 
a  ,  b  .  c i  ,  k  »  l^ 

having   n    terms,    and  the   common   difference   d,   and   designate 
the   term  which  has  p   terms  before   it,  by  Z,  we   shall   have 

t  =  a+pd (1). 

If  we  revert  the  order  of  terms  of  the  progression,  con- 
eidering  I  as  the  first  term,  we  shall  have  a  new  progression 
whose  common  difference  is  —  cZ.  The  term  of  this  new  pro- 
gression which  has  p  terms  before  it,  will  evidently  be  the  same 
as  that  which  has  p  terms  after  it  in  the  given  progression, 
and  if  we   represent   that  term   by  t\  we   shall  have, 

t'z=zl-pd   -  ...     (2). 


CHAP,   yill.j  ARIIHMETICAL  PROGRESSIOlSr.  237 

Adding   equations  (1)  and  (2),  member  to  member,  we  find 
i  +  t^  z=:  a  +  I  'y     hence, 

The  sum  of  any  two  terras^  at  equal  distances  from  the  extremes 
of  an  arithmetical  progression^  is  equal  to  the  sum  of  the  extremes, 

176.  If  the  sum  of  the  terms  of  a  progression  be  repre- 
sented by  S^  and  a  new  progression  be  formed,  by  reversing 
ih?   order    of  the   terms,  we   shall   have 

S=za  +  h  +  c+  .  .  .  .  +i  +  Jc+l, 
S=l  +  1c  +  i+....+c-\-b  +  a, 
Adding   these   equations,  member   to   member,  we   get 
2S={a+l)+{h  +  Jc)+{c  +  i)...  +(i  +  c)  +  {k  +  h)  +  {l+a)', 

and,  since  all  the  sums,  a  +  ^,  h  -\-  k^  c  +  t  .  .  .  .  are  equal 
to  each  other,  and  their  number  equal  to  w,  the  number  of 
^erms   in   the   progression,  we   have 

2ASf  =  (a  +  I)  n,      or      8=  (~4~~)  ^  '     ^^^^  ^^' 

The  sum  of  the  terms  of  an  arithmetical  progression  is  equal  to 
half  the  sum  of  the  two  extremes  multiplied  by  the  number  of  terms, 

EXAMPLES. 

1.  The  extremes  are  2  and  16,  and  the  number  of  terms  8: 
fv'hat   is   the   sum   of  the   series  ? 

S=\—^\xn,      gives      >S^  =  — - — x  8  =  72. 

2.  The  extremes  are  3  and  27,  and  the  number  of  terms  12  * 
what   is   the   sum   of  the   series  ?  Ans,  180. 

3.  The  extremes  are  4  and  20,  and  the  number  of  terms  10: 
what  is   the   sum   cf  the   series  ?  Ans,  120. 

4.  The  extremes  arc  8  and  80,  and  the  number  of  terms  10: 
what  is  the   sum  of  the  series  1  Ans,  440. 

The  formulas 

l  =  a  +  {n^\)d      and     ^f=/^)<w, 


238 


ELEMENTS   OF  ALGEBBA. 


[CHAP.   VIIL 


contain  five  quantities,   a,  d^  n,  I,  and  S,  and   consequently  give 
rise   to   the   following  general  problem,  viz. : 

Anj/  three  of  these  Jive  quantities  being  given,  to  determine  the 
other  two. 

This  general  problem  gives  rise  to  the  ten  following  cases  : — 


No.     Given.   fJnknown. 


Values  of  the  Unknown  CluaAitities. 


a,  d,  n 


L  S 


lz=za-{-{n  —  l)d',   S=zin[2a-{-{n  —  l)d]. 


r^,  S 


^  — «   .    -.        ^       (1+  a)(l  —  a  +  d) 


a,d,S 


n,  I 


d—2a±  ^{d—2a)^-{-SdS 


2d 


;  I  =  a -i- {n  —  l)d. 


a,  n,  I 


S,  d 


Sz=zin{a-{-l)',    d  = 


I -a 


a^n,  S 


d,  I 


2(S-  an)       ,        2S 

f TV  5    ^  = ^• 

n(n  —  1 )  n 


a,  l,S 


n,  d 


2S       ^_{l-{-a){l-a) 
,]  d- 


a-\-V  2^-(/  +  a) 


d^  n,  I 


«,  S 


a=zl-{n-l)d',    S=:in[2l-{n-l)d], 


d,  n,  S 


a,  / 


2^- 


n  {n-l)d       _  2S+n{n~l)d 


2n 


2n 


d,l,S 


2l-\-d:h^{2l-{-d)^-SdS\ 
2d  ' 


:l—(^ri-^l)d. 


10 


n,  I,  S 


a,  d 


2S 


■I:    d  = 


2  (ill  -  S) 
n(7i  —  \)' 


177.  From   the  formula 

I  z=z  a  -{-  (n  —  V)  d, 
we  have,  a  =.  I  —  (n  —  l)o?;     that   is. 

The  first  term  of  an  increasing  arithmetical  progression^  is  equal 

to  any  following  term,  minus  the  product  of  the  common  difference 

by   the  number  of  preceding  terms, 

178,  From   the   same   formula,  we   also   find 

^  —<^ 

d  z=z ;     that  is, 

7i  —  1  ' 


CHAP.    VIII.J  ARITHMETICAL   PKOGKESSION.  239 

In  any  arithmetical  progression^  the  common  difference  is  equal 
to  the  last  term  minus  the  first  term^  divided  by  the  number  of 
terms   less   one. 

If  the  last  term  is  less  than  the  first,  the  common  diflerenco 
Will   be   negative,  as   it   should   be. 

EXAMPLES. 

1.  The  first  term  of  a  progression  is  4  the  last  term  16,  and 
the  number  of  terms  considered  5  :  what  is  the  common 
difference  ? 

The   formula 

I  — a 

2.  The  first  term  of  a  progression  is  22,  the  last  term  4, 
and  the  number  of  terms  considered  10 :  what  is  the  common 
difference  ?  Ans,  —  2. 

179.  By  the  aid  of  the  last  principle  deduced,  we  can  solve 
the   following  problem,  viz. : 

To  find  a  number  m  of  arithmetical  means  between  two  g^veii 
numbers  a   and   b. 

To  solve  this  problem,  it  is  first  necessary  to  find  the  com- 
mon difference.  Now,  we  may  regard  a  as  the  first  term  of 
an  arithmetical  progression,  b  as  the  last  term,  and  the  required 
means  as  intermediate  terms.  The  number  of  terms  considered, 
of  this  progression,  will  be  expressed  by  m  +  2, 

Now,  by  substituting  in  the  above  formula,  b  for  /,  and  m  +  2 
for  n,  it  becomes 

-  b  —  a  .       b  --  a 

that  is,  the  common  difference  of  the  required  progression  Is 
obtained  by  dividing  the  difference  between  the  last  and  first 
terms  by  one   more  than   the  required  number  of  means. 


240  ELEMENTS   OF  ALGEBr\.  [CHAP.   VIII, 

Having  obtained  the  common  diiference,  form  the  second  term 

of  the  progression,  or  the  first  arithmetical  mean^  by  adding  g?,  or 

J  

-,    to   the  first   term   a.     The   second  mean   is   obtained  by 

augmenting   the  first  by  c?,  &c. 

EXAMPLES. 

1.  Find  3  arithmetical  means  between  2  and  18.     The  formula 

^      b-a         .  18-2       , 

d  =  — — -,      gives     d  =  — =  4  ; 

m+1       °  4 

hence,  the   progression  is 

2  .  6  .  10  .  14  .  18. 

2.  Find    12   arithmetical    means  between  77   and   12.      The 

formula 

b  -a        ,  ,      12-77 

__,     g,ves    d  =  —^^=^5; 

lience,  the  progression  is 

77  .  72  .  67  .  62 22  .  17  .  12. 

3.  Find  9  arithmetical  means  and  the  series,  between  75 
and  5. 

Ans.    Progression  75  .  68  .  61 26  .  19  .  12  .  5. 

ISO.  If  the  same  number  of  arithmetical  means  be  inserted 
between  the  terms  of  a  progression,  taken  two  and  two,  these 
terms,  and  the  arithmetical  means  together,  will  form  one  and 
the   same   progression. 

For,  let  a.b,c.e.f.,,.he  the  proposed  progression, 
aiid  m  the  number  of  means  to  be  inserted  between  a  and  i, 
h  and  c,   c  and  e 

From  what  has    just    been    said,   the   common  difference  of 
each  partial  progression  will  be   expressed  by 
b  —  a        c  ~-  b        e  —  c 
m+V      ^hFI'      nTfl  '  '  '  ' 
which  are  equal  to  each   other,   since,   a,   5,   c,  .  .  .  are  in  pro 
gression:   therefore,  the  common  difference  is  the   same  in  each 


CHAP.   VIII.J  AEITHMETICAL  PROGRESSION.  241 

of  the  partial  progressions ;  and  since  the  last  term  of  t^ie  first, 
forms  the  Jirst  terra  of  the  second,  &c.,  we  may  conclude  that 
all   of  these  partial  progressions  form  a  single  progression. 

GENERAL    EXAMPLES. 

1.  Find  the  sum  of  the  first  fifty  terms  of  the  progression 

2  .  9  .  16  .  23  .  .  . 

For  the  50th  term,   we  have  \ 

Z  =  2  +  49  X  7  =  345. 

50 
Hence,         /S  =  (2  +  345)  x  —  =  347  x  25  =  8675. 

2.  Find   the   100th   term  of  the    series   2  .  9  .  16  .  23  .  . 

Ans.    695. 

3.  Find  the  sum  of  100  terms  of  the  series  1.3.5.7.9... 

Ans,    10000. 

4.  The  greatest  term  considered  is  70,  the  common  difference 
3,  and  the  number  of  terms  21 :  what  is  the  least  term  and 
the   sum  of  the   terms'? 

Ans,    Least  term  10 ;   sum  of  terms  840. 

5.  The  first  term  of  a  decreasing  arithmetical  progression  is 
10,  the  common  difference  is  —  ^-,  and  the  number  of  terms 
21  :   required   the   sum  of  the   terms.  Ans,    140. 

6.  In  a  progression  by  differences,  having  given  the  common 
difference  6,  the  last  term  185,  and  the  sum  of  the  terms  2945 : 
And  the  first  term,  and  the  number  of  terms. 

Ans.    First   term  =5;   number  of  terms  3L 

7.  Find  9  arithmetical  means  between  each  antecedent  and 
consequent  of  the   progression   2. 5. 8.  11.  14  .  .  . 

Ans.   dz=:0,X    ' 

8.  Find  the  number  of  men  contained  in  a  triangular  bat- 
talion, the  first  rank  containing  1  man,  the  second  2,  the  third 
3,   and   so   on   to    the   w*^,   which   contains   n.     In    other  words, 

16 


242  ELEMENTS   OF  ALGEBRA.  [CHAP.   VIII. 

find   the   expression   for   the   sum  of  the   natural  numbers  1,  2, 
3,   .  .  .   from   1    to   n.   inclusively.  ,      ,    ^^ 

Ans.   ^  =  "("+^>. 

At 

9.  Find  the  sum  of  the  first  n  terncs  of  the  progression  of 
uneven  numbers    1,    3^   5,   7,    9  .  .  .  Ans,    S  =z  n\ 

10.  One  hundred  stones  being  placed  on  the  ground,  in  a 
straight  line,  at  the  distance  of  two  yards  from  each  other,  how 
far  will  a  person  travel  who  shall  bring  them  one  by  one  to 
a   basket,  placed   at   two   yards   from   the   first  stone] 

Ans.    11    miles    840  yards. 

Of  Ratio  and  Geometrical  Proportion. 

181.  The  Eatio  of  one  quantity  to  another,  is  the  quotient 
which  arises  from  dividing  the   second    by   the    first.      Thus,  the 

ratio  of  a  to  5,  is   — . 
a 

182.  Ttoo  quantities  are  said  to  be  proportional,  or  in  pro- 
portion, w^hen  their  ratio  remains  the  same,  while  the  quantities 
themselves  undergo  changes  of  value.  Thus,  if  the  ratio  of  a 
to  h  remains  the  same,  while  a  and  h  undergo  changes  of  value, 
then  a  is   said   to   be   proportional   to   6. 

183«  Four  quantities  are  in  proportions  when  the  ratio  of  the 
first  to  the  second,  is  equal  to  the  ratio  of  the  third  to  the 
fourth. 

Thus,   if 

L-  — 
a         c  ^ 

the  quantities  a,  5,  c  and  c?,  are  said  to  be  in  propoition.  We 
generally  express  that  these  quantities  are  proportional  by  wrilii  g 
them  as  follows  : 

a  \  h  \    :  c  :  d. 

This  algebraic  expression  is  read,  a  is  to  h,  as  ib  to  J, 
and   is   called  a  proportion. 


CHAP.   VIII.]  GEOMETRICAL   PROGRESSION"  243 

184.  The  quantities  compared,  are  called  terms  of  the  pro- 
portion. 

Tlie  first  and  fourth  teri.is  are  called  the  extremes^  the  seconci 
and  third  are  called  the  means ;  the  first  and  third  are  called 
ontecedents^  the  second  and  fourth  are  called  consequents^  and  the 
fourth  is  said  to  be  a  fourth  proportional  to  the  other  three. 

If  the  second  and  third  terms  are  the  same,  either  of  these 
is  said  to  be  a  mean  proportional  between  the  other  two.  Thus, 
in  the  proportion 

a  :  b  :    :  b  :  c^ 
6  is  a  mean  proportional   between   a  and   c,  and  c  is  said  to  be 
a  third  proportional  to  a  and  b. 

185t  Two  quantities  are  reciprocally  proportional  when  one  is 
proportional  to  the  reciprocal  of  the  other. 

Geometrical  Progression. 

186»  A  Geometrical  Progression  is  a  series  of  terms,  each 
of  which  is  derived  from  the  preceding  one,  by  multiplying  it 
Sy  a  constant  quantity,  called  the  ratio  of  the  progression. 

If  the  ratio  is  greater  than  1,  each  term  is  greater  than  it\e 
preceding  one,  and  the  progression  is  said  to  be  increasing^ 

If  tlie  ratio  is  less  than  1,  each  term  is  less  than  the  pn^ 
ceding  one,  and  the  progression   is   said   to   be  decreasing. 

Thus, 
...    3,    6,  12,  24,  .  .  .  &c.,  is  an  increasing  progression. 

...  16,  8,    4,     2,  1,   — ,   — ,  ...  is  a  decreasing  progressi^ni 

It  may  be  observed  that  a  geometrical  progression  is  a  con- 
tinued proportion  in  which  each  term  is  a  mean  proportions^ 
between  the  preceding  and  succeeding  terms. 

187.  Let  r  designate  the   ratio  of  a   geometrical  progression, 

a  :  6  :  c  :  c?,  .  .  .  .  &c. 
We  deduce  from   the   definition   of  a   progression  the  follow 
ing   equations : 

b  =  ar,     c  =  br  zzz  ar'^\,     d  z=  cr  zznar^^     e  =1  dr  :==.  ar^      .  . ; 


244  ELEMENTS   OF  ALGEBRA*.  [CHAP.    VIIL 

and,  Ji  geieral,  anj  term  w,  that  is,  one  which  has  n  —  I  terms 
before   it,  is  expressed  bj  ar**~i. 

Let  I  be  this  term ;  we  have  the  formula 

by    means   of   which   we    can    obtain    any   term   without    being 
obliged    to  find   all   the  terms  which  precede   it.     That  is, 

An^  term  of  a  geometrical  progression  is  equal  to  the  first  term 
multiplied  hy  the  ratio  raised  to  a  power  whose  exponent  denotes 
the  number  of  preceding  terms, 

EXAMPLES. 

1.  Find  the  5th  term  of  the  progression 

2  :  4  :  8  :  16,  &c., 
in  which   the  first  term  is  2,  and  the   common  ratio   2. 
5th  term  =  2  X  2*  =  2  X  16  =  32. 

2.  Find  the   8th  term  of  the  progression 

2  :  6  :  18  :  54  .  .  . 
8th  term  =  2  X  3^  =  2  X  2187  =  4374. 

3.  Find  the  12th  term  of  the  progression 

1  ^ 

64  :  16  :  4  :  1  :  4-  .  . 
4 


/ 1  v^     43      1 

12th  term  =  64  (-)    =^=^3: 


65536 


188.  We  will  now  explain  the  method  of  determining  the  sum 
of  n  terms  of  the  progression 

a  :  h  \  c  '.  d  I  e  \  f  :  ,  ,  »  I  i  ',  k  ',  l^ 
of  which   the   ratio   is   r. 

If  we   denote  the   sum  of  the  series  by  S^  and  the  n'*  leim 
Dy  I-  we  shall  have 

aS^  =  a  +  ar  +  ar*-*  .  .  .  .  +  ar^"^  +  ar^^^^ 
If  we   multiply  b:)th   members  by  r,  we  have 

Sr  zzzar  -\-  ar^  +  ar^  ,  ,  ,  +  ar^"-  +  ar*  ; 


CHAP.   VIII.l  GEOMETRICAL  PROGRESSION.  246 

and  by  subtracting  the  first  equation  from  the  second,  member 
from  member, 

-,      ctr^  —  a 
Sr  —  S  =:ar'^  —  a,       whence,       S  = r  \ 

substituting  for  ar",  its  value  /r,  we   have 

^  — .     that  IS, 

r  —  1 

To  obtain  the  sum  of  any  number  of  terms  of  a  progression 
by  quotients. 

Multiply  the  last  term  by  the  ratio,  subtract  the  first  term  from 
this  product,  and  divide  the  remainder  by  the  ratio  diminished  by  1. 

EXAMPLES. 

1.  Find   the   sum  of  eight   terms   of  the   progression 

2  :  6  ;  18  :  54  :  162  ...  :  4374. 

^^^a       13122^-2 
r-l  2 

2.  Find   the   sum   of  five   terms  of  the   progression 

2  :  4  :  8  :  16  :  32;  .  .  .  . 

^=^  =  ^^=62. 
r  —  1  1 

3.  Find   the   sum   of  ten   terms   of  the   progression 

2  :  6  :  18  :  54  :  162  ...  2  X  39  =  39366. 

Ans.  59048. 

4.  What  debt  may  be  discharged  in  a  year,  or  twelve  months, 
by  paying  $1  the  first  month,  |2  the  second  month,  $4  the  third 
month,  and  so  on,  each  succeeding  payment  being  double  tho 
last ;  and  what  will   be  the  last  payment  ? 

Ans,  Debt,  44095  ;  last  payment,  $2048. 

5.  A  gentleman  married  his  daughter  on  New- Year's  day,  and 
gave  her  husband  Is.  toward  her  portion,  and  was  to  double  it 
on  the  first  day  of  every  month  during  the  year :  what  was  hei 
portion '2  Ans.  £204  Vos. 


246  ELEMENTS   OF   ALGEBRjT.  [CHAP.    VIII. 

6.  A  man  bought  10  bushels  of  wheat  on  the  condition  that 
he  should  pay  1  cent  for  the  first  bushel,  3  for  the  second,  9 
foi  the  third,  and  so  on  to  the  last :  what  did  he  pay  for 
the  last   bushel,  and  for   the   ten   bushels  ? 

Ans,  Last  bushel,  $196  83 ;  total  cost,  $295,  24. 

189.  When  the  progression  is  decreasing,  we  have  r  <  1  and 
/  <  a ;  the  above  formula  for  the  sum  is  then  written  under 
the   form 

in   order   that  both   terms  of  the  fraction  may  be  positive. 
By  substituting   ar^-^   for  /,  in    the   expression   for  S, 

^       ar^  —  a  ^     a  —  ar^ 

o  = -r-.      or      o  =  - 


1  1  —  r 

EXAMPLES. 

1.  Find   the   sum   of  the  first  five   terms   of  the   progression 

32  :  16  :  8  :  4  :  2. 

32  -  2  X  4-       oi 

a  —  lr       2        31 

S  —  = =  —  =  62.  f 

\  —  r  1  1 

2.  Find  the   sum  of  the  first  twelve  terms  of  the  progression 

1  1 


64  :  16  :  4  :  1 


4 65536' 


64  -  --^—  X  4-      256  ^ 


^     a -It  65536        4  65536      ^,         65535 

8  =- = —  =  85  -f 


X-r  ^  3      -        '    196608* 

4 

We  perceive  that  the  principal  difficulty  consists  in  obtaining 
ihe  numerical  value  of  the  last  term,  a  tedious  operation,  even 
whon    the    number  of  terms   is   not  very  great. 

190.    If  in   the   formula 

a(T^  -  1) 


CHAP.   VIII.]  GEOMETRICAL  PROGRESSION.  247 

i»e  make  ri=l,   it  reduces  to 

^-  0- 

This  result  sometimes  indicates  indetermination ;  but  it  often 
arises  from  the  existence  of  a  common  factor  in  both  numerator 
and  denominator  of  the  fraction,  which  factor  becomes  0,  in  con* 
sequence  of  a  particular  supposition. 

Such  is  the  fact  in  the  present  case,  since  both  terms  of  the 
fraction  contain  the  factor  r  —  1,  which  becomes  0,  for  the  par- 
ticular  supposition   r  =  1, 

If  we  divide  both  terms  of  the  fraction  by  this  common  factor, 
we  shall  find  (Art.  60), 

S  =  ar"^-^  +  ar'^-'^  +  ar'^-'^  +   ....   +  ar  +  a, 

in  which,  if  we  make  r  =  1,  we  get 

/S=ia+a  +  a+«+ +a=  na. 

We  ought  to  have  obtained  this  result;  for,  under  the  suppo- 
sition made,  each  term  of  the  progression  became  equal  to  a, 
a^d  since  there  are  n  of  them,  their  sum  should  be  na, 

191  •  From  the  two  formulas 

r^       l^  —  « 

I  =  ar«-\     and     S  = , 

r  —  1  ,^ 

several    properties   may   be    deduced.      We  shall   consider   only 

some  of  the  most  important. 

The   first  formula   gives 

I      ,  ^-^  rr 

r^~^  z=: —       whence      r=      \/  — . 
a  \    a 

The   expression 

_  «-i   FT 

V    a' 

furnishes  the  means  for  resolving  the  following  problem,  viz . 

To  find  m  geometrical  means  between  two  given  numbers  a  and 
b ;  that  is,  to  find  a  number  m  of  means,  which  will  form  with  a 
and  I,  considered  as  extremes,  a  geometrical  progression. 


248  ELEMENTS   OF  ALGEBRA.  imAP.   VIII, 

To  find  this  series,  it  is  only  necessary  to  know  the  ratio. 
Now,  the  required  number  of  means  being  m,  the  total  number 
of  terms  considered,  will  be  equal  to  m  -f  2.  Moreover,  we 
hare  I  z=zb;  therefore,  the  value  of  r  becomes 

r  =:     \/  —  ;    that  is, 

To  find  the    ratio^  divide  the  second    of  the  giveii  numbers   by  the 
first;    then    extract    that  root  of  the    quotient   whose    index  is    one 
(jreater    than    the   required   number  of  means : 
Hence  the  progression  is 


a  :  a    \/  —  :  a    \/  —::  :  a    \/  — r: :  .  .  .  5. 
a 


EXAMPLES. 

1.  To  insert  six  geometrical  means  between  the  numbers  3 
and  384,  we  make  m  =  Q,   whence   from   the   formula, 

hence,  we  deduce   the   progression 

3  :  6  :  12  :  24  :  48  :  96  :  192  :  384.       ' 

2.  Insert  four  geometrical  means  between  the  numbers  2  and 
486.     The  progression  is 

2  :  6  :  18  :  54  :  162  :  486. 

ReMxVrk. — When  the  same  number  of  geometrical  means  are 
inserted  between  each  two  of  the  terms  of  a  geometrical  pro- 
gression, all  the  progressions  thus  formed  will,  when  ^aken  to- 
gether, constitute  a  single  progression. 

Progressions  having  an  infinite  number  of  terms, 

192t  Let  there  be  the  decreasing  progression 

a  :  b  :  c  :  d  :  e  :  f  '^  c  .  ., 

containing  an  infinite  number  of  terms.     The  formula 

%  —  ar^ 
^_     

I  —  r 


CHAP.    VIII.]  GEOMETRICAL   PROGRESSIOJS-.  249 

which  expresses  the  sum  of  n  terms,  can  be  put  under  the  form 


1  —  r       1  —  r* 

Now,  since  the  progression  is  decreasing,  r  is  a  proper  frac- 
tion, and  r"  is  also  a  fraction,  which  diminishes  as  n  increases. 
Therefore,  the  greater  the  number  of  teims   we  take,  the  more 

will X  r^  diminish,  and  consequently,  the  nearer  will  the 

1  —  r 

sum   of  these   terms  approximate  to   an  equality  with  the  first 

part  of  S ;  that  is,  to .      Finally,  when  n  is  taken  greater 

than  any  assignable  number,  or  when 

n  =:  cx>,     then X  r^ 

will   be  less  than  any  assignable  number,  or  will  become  equal 
to  0 ;   and  the  expression will  represent  the  true  value  of 

the  sum  of  all  the  terms  of  the  series.     Hence, 

The  sum  of  the  terms  of  a  decreasing  progression^  in  which  the 
number  of  terms  is  infinite^  is 


1  -  / 

This  is,  properly  speaking,  the  limit  to  which  the  partial  sums 
approach,  as  we  take  a  greater  number  of  terms  of  the  pro- 
gression.    The  number  of  terms   may   be   taken   so   great  as  to 

make  the  difference  between   the  sum,  and ,    as  small  as 

1  —  r 

we  please,  and  the   difference  will  only  become  zero  ^hen    the 

number  of  terms  taken  is  infinite. 

EXAMPLES. 

1.  Find  the  sum  of 

1      1      1       1       1       . 


250  ELEMENTS   OF   ALGEBRI.  [CHAP.   VEIL 

We  have    for  the  sum  of  the  terms, 

1  3 


S  = 


1  -  r  ~       J[_  "   2  • 
3 


2,  Again,  take  the  progression 

1     1     1       1       1       1       . 
•   2  *•  4  '  8  '  16  '  32  '   '^''-  •  •  • 

We  have  S  =  —^^  =  -^  =  2. 

1  —  r  1 

What  is  the  error,  in  each  example  for  ?i  =  4,  n  =  5,  ^  =  61 

Indeterminate    Co-efficients. 

193.  An  Identical  Equation  is  one  which  is  satisfied  for  any 
values  that  may  be  assigned  to  one  or  more  of  the  quantities 
which   enter  it.     It  differs  materially  from  an  ordinary  equation. 

The  latter,  when  it  contains  but  one  unknown  quantity,  can 
only  be  satisfied  for  a  limited  number  of  values  of  that  quan- 
tity,  whilst  the  former  is  satisfied  for  any  value  whatever  of 
the   indeterminate  quantity  which  enters  it. 

It  differs  also  from  the  indeterminate  equation.  Thus,  if  in 
the  ordinary  equation 

ax  -i-  hj  -\-  cz  -}-  d  z=  0 

values   be  assigned  to   x  and  y  at  pleasure,  and  corresponding 
values  of  z  be   deduced  from   the   equation,  these   values   taken 
together  will    satisfy   the   equation,    and   an   infinite   number   of 
sets   of  values  may  be  found  which  will  satisfy  it  (Art.  88). 
But   if  in   the   equation 

ax  -{-  by  +  cz  +  d  =  0, 
we   impose    the    condition    that    it    shall    be    satisfied  for    any 
valies  of  X,   y   and   2?,  taken   at  pleasure,  it  is  then  called  an 
identical  equation, 

194.  A  quantity  is  indeterminate  when  it  admits  of  an  infinite 
number  of  values. 

Let  us  assume  the  identical  equation, 

A  +  Bx  4-  Cu:^  -j-  i>2;3  +  &c  =  :?    -    -    -    -    (1), 


CHAP.   VIII.]  GEOMETRICAL   PROGRESSION.  251 

in   which    the   co-efficients,   A,  B,   C,  D,  &;c.,  are   entirely   inde 
pendent  of  x,  * 

If  we  make  a?  =  0  in  equation  (1)  all  the  termr  containing 
X   reduce   to   0,    and   we   find 

^  =  0. 
Substituting    this    value   of  A   in    equation  (1).  and   factoring, 
it   becomes, 

x{B+  Cx  +  Dx^  +  &c.,)  =  0 (2), 

which   may  be   satisfied   by  placing   a;  =  0,    or   by   placing 

B-\-  Cx  +  Bx^  ^  k.(^,^^ (3). 

The  first  supposition  gives  a  common  equation,  satisfied  only 
for  a;  =  0.  Hence,  equation  (2)  can  only  be  an  identical  equa- 
tion  under  a  supposition  which  makes  equation  (3)  an  identical 
equation. 

If,  now,  we  make  a;  =  0  in  equation  (3),  all  the  terms  con- 
taining  X  will  reduce  to  0,  and  we  find 

^  =  0. 
Substituting   this   value    of  B  in    equation    (3),  and  factoring, 
we  get 

a:((7+i>^+  &c.)  =  0 (4). 

In  the  same  manner  as  before,  we  may  show  that  (7=0, 
and  so  we  may  prove  in  succession  that  each  of  the  co-efficients 
i>,  E^  &c.,  is   separately  equal  to  0 :   hence. 

In  every  identical  equation,  either  member  of  which  is  0,  in- 
volving a  single  indeterminate  quantity^  the  co-efficients  of  tlie 
different  powers  of  this   quantity   are   separately  equal   to   0. 

195i  Let   us   next  assume  the  identical   equation 

a  +  bx  +  cx^  +  &c.  z=  a'  +  b'x  +  c'x'^  +  &c. 
By  transposing  all   the  terms  into   the  first  member,  it  may 
be   placed  under   the   form 

(a  -  a')  +  {b~b')x  +  {c  —  ^')  x^  +  (S^c.  =  0. 
Now,  from   the   principle  just   demonstrated,  ^  ^^ 

a-a'=;cO,    5  — 6'  — 0,    c  -  c' =  0,    k>Q^^(t,^  ^ 
whence  a  =  a'  ,  6  =  6'    c  =  c'  ,  &;o.,  &c. ;     that  is, 


252  ELEMENTS   OF  ALGEBRA.  [CHAP.    VIU. 

In  an  identical  equation  containing  hut  one  indeterminate  quan- 
tity,  the  co-efficients  of  the  like  powers  of  that  quantity  in  the 
two  members,   are   equal    to   each   other. 

196»  We  may  extend  the  principles  just  deduced  to  identical 
equations   containing   any  number   of  indeterminate  quantities. 
T'or,  let   us   assume   that   the   equation 
a  +  hx  +  h'y  +  b"z  +  &c.  +  cx^  +  c'y^  +  c"z^  +  &c.  f  dx^ 
+  dy  +  &c.  =  0     -     -     .     (1), 
is   satisfied   independently   of  any  values   that   may   be   assigned 
to   X,  y,  z,  &;c.     If    we    make    all   the   indeterminate    quantities 
xcept  X   equal   to   0,  equation  (1),  will   reduce   to 

a  +  bx  +  cx"^  -i-  dx^  +  &;c.  =  0  ; 

whence,  from   the   principle  of  article    194, 

a  =  0,     b=zO,     c  =  0,     d=zO,     &cc.- 0 

If,  now,  we  make  all  the  arbitrary  quantities  except  y  equal 
to   0,  equation  (1)  reduces  to, 

L  +  b^y  +  cY  +  dy^  +  &cr=  0 ; 
whence,  as  before^  J 

a  =  0,     b'  =  0,     c'  =  0,     d'  =  0,     &c.  Z^(^ 
and   similarly  we   have 

b"  =  0,     c''  =  O;     &c.  J 

The  principle  here  developed  is  called  the  principle  of  inde 
terminate  co-efficients^  not  because  the  co-efficients  are  really 
indeterminate,  for  we  have  shown  that  they  are  separately 
equal  to  0,  but  because  they  are  co-efficients  of  indeterminate 
quantities. 

197t  The  principle  of  Indeterminate  Co-efficients  is  much  used 
in  developing   algebraic   expressions  into   series. 

For  example,  let   us   endeavor   to  develop  the   expression, 

a 

a'  -{-h'x' 

into  a   series  arranged   according   to  the  ascending  powers  of  a?* 


CHAP.   yilL]  GEOMETEICAL  PROGRESSION.  253 

Let   us   assume   a   development    of  the   proposed  form, 

-r^--=P+Qx  +  Rx^  +  Sx^-\-^(t,     -    -     -     (1), 
a  -{-ox 

in  which  P,   Q^  M,  &c.,  are  independent  of  x,  and  depend  upon 

a,  a^  and  5'  for   their  values.     It  is   now  required    to   find   such 

values  for  P,   Q^  B,  &;c.,  as  will  make  the   development  a  true 

one  for  all   values   of  x. 

By  clearing  of  fractions  and  transposing  all  the  terms  into 
the   first   member,  we   have 

Fa'    -\-  Qa'   x -\-  Ra'    x^  +  &c.  =  0. 
—  a-{-FW      +   qV  &c. 

Since   this   equation  is   true  for   all   values   of  a:,  it   is  identi- 
cal,   and   from   the   principle  of  Art.   194,  we  have 
Pa'  —a-  0,    Qa'  +  Ph'  =  0,    Pa'  +  Qb'  =  0,  &c.,  &c. ;  whence, 

^      a      ^            Pb'  ab'     ^  Qb'       ab"^     , 

a'                     a  a'^  a  a'-^ 

Substituting  these  values  of  P,  Q^  P,  &;c.,  in  equation  (1), 
it  becomes    .            . 

7T6^=5-S'^  +  ^^'-Sr'^^  +  *^«-     -     -     (2). 

Since  we  may  pursue  th§  same  course  of  reasoning  upon 
any  like  expression,  we  have  for  developing  an  algebraic  ex- 
pression  into  a   series,  the  following 

RULE.  . 

I.  Place  the  given  expression  equal  to  a  development  of  the 
form  P  +  Qx  +  Px^  +  do.,  clear  the  resulting  equation  of  frac- 
tions, and  transpose  all  of  the  term^s  into  the  first  member  of 
the   equation, 

II.  Then  place  the  co-efficients  of  the  different  powers  cf  the  let* 
ter,  with  reference  to  which  the  series  is  arranged,  separately  equal 
to   0,  and  from    these  equations  find   the  values  of  P,    Q^  P,  c£r. 

III.  Having  found  these  values,  substitute  them  for  P,  Q,  R,  &c.^ 
in  the  assumed  development,  a-^d  the  result  will  be  the  develop* 
ment  required. 


254   '  ELEMENTS   OF   ALGEBftA.  [CHAP.   VIIT, 


EXAMPLES. 


1.  Develop     into  a  series. 


CC  X  X 


2.  Develop    '-. ^     into   a  series. 

^      (a  —  xy 

tt^        a-^        a*  a^ 

„    ^      ,         1  4-  2.t?    . 

'  3.  Develop —     into  a  series. 

J  —  ox 

Ans.   1  +  5a:  +  15^-2  +  45:^3  .|.  135^4  ^  &c. 

198o  We  have  hitherto  supposed  the  series  to  be  arranged 
according  to  the  ascending  powers  of  the  unknown  quantity, 
commencing  with  the  0  power,  but  all  expressions  cannot  b(: 
developed  according  to  this  law.  In  such  cases,  the  application 
of  the  rule  gives  rise  to  some  absurdity. 

For   example,   if  we   apply    the   rule   to   develop -^  we 

shall   have, 

1 


:  P  +    §^  +  i^^^  +  &C.    -    -    -     (1). 


*6x  —  x"^ 
Clearing  of  fractions,  and  transposing, 

-l  +  3P^  +  3§    a:2+&c.  =0; 
-  P 
Whence,  by  the  rule, 

-1=0,     3P  =  0,     3§-P=:0,  &c. 

Now,  the  first  equation  is  absurd,  since  —  1  cannot  equal  0. 
llence,  we  conclude  that  the  expression  cannot  be  developed  ao 
cording  to  the  ascending  powers  of  x^  beginning  at  x^. 

We    may,    however,    write    the    expression    under    tlie    forii- 

—  X ,  and  by  the  application  of  the  rule,  develop  tjie  fact^ 

x        3  —  a; 

,   which   gives 

3^  =  T^-¥"  ^27'^+8l*   +*^<^- 


CHAP.   VIII.J  RECURRING  SERIES.  255 

whence,  by  substitution, 


Sx-x^       Sx  '     9        '27      '81 

Since  —  is  equal  to  Sx-^  (Art.  166),  we  see  that  the  true  devel- 
opment contains  a  term  with  a  negative  exponent,  and  the  sup- 
position made  in  equation  (1)  ought  to  have  failed. 

Recurring  Series. 

199.  The  development  of  fractions  of  the   form  ■     .   ,.  ,   &c., 

a-\-bx 

gives  rise  to  the  consideration  of  a  kind  of  series,  called  recur- 
ring  series. 

A  HEcuRRiNG  SERIES  is  onc  in  which  any  term  is  equal  to  the 
algebraic  sum  of  the  products  obtained  by  multiplying  one  or 
more  of  the   preceding  terms  by  certain  fixed  quantities. 

These  fixed  quantities,  taken  in  their  proper  order,  constitute 
what  is  called  the  scale  of  the  series, 

200.  If  we  examine  the  development 

a  a         ah'  ah'"^  ah'^ 

a  -[-  h  X        a'        a  2  a'^  a^ 

we  shall  see,  that  each  term  is  formed  by  multiplying  the  pre- 
ceding one  by jx.     This  is  called  a   recurring    series  of  the 

first  order ^  because  the  scale  of  the  series  contains  but  one 
term. 

The  expression ;-ar  is  the   scale   of  ike  series,  and  the   ex 

pression ^  is  called  the  scale  of  the  co-efficients. 

It  may  be  remarked,  that  a  geometrical  progression  is  a  recur 
ring  series  of  the  first  order.  ^-^ 

201.  Let  it  be  required  to  develop  the  expression 

a  -\-  hx 


'2'  +b'x+c'x^ 


into  a  series. 


256 


Assume 


ELEMENTS   OF   ALGEBRA. 
a  -\-  bx 


a'  -{•  b'x  -f-  c'a;2 
Clearing  of  fractions,  and  transposing,  we  get 


[CHAP.  VIIL 
P-\-  Qx^  Re''  +  Sx^  +  &C 


Pa' 

+  Qa' 

X  +  Pa' 

x^  +  Sa' 

—  a 

+  Ph' 

+  Qb' 

+  Rb' 

—  b 

+  Pc^ 

+  Qc' 

Therefore,  we  have 

Pa'  -     a  =  0, 

Qa'  +  Pb'  -^     6  =  0, 

Ba'  +  QV+  P&  =  0^ 

Sa'  -hpy+  Q</=0, 
&c.,     &;c.. 


x^  +  &c.  =  0. 


a'  a' 

(fee.,     (See.; 


from   which  we   see  that,  commencing  at   the   third,  each  co-effi- 
cient is   formed  by  multiplying   the   two   which  precede   it,   re- 

spectively,    by j  and ^,    viz.,    that    which    immediately 

b' 

precedes  the  required  co-efficient  by 7,    that   which   precedes 


it  two  terms  by 7,  and  taking  the  algebraic  sum  of  the  pro 

ducts.     Hence, 

\       a''  a'} 

is  the  scale  of  the  co-efficients, 

From  this  law  of  formation  of  the  co-efficients,  it  follows  that 

the  third  term,  and   every   succeeding   one,  is  formed  by  multi- 

b' 
plying  the  one  that  next  precedes  it  by ^ar,  and   the  second 

preceding  one   by ;  a;^,  and  then  taking  the  algebraic  sum  of 

these  products :   hence, 

is  the  scales  of  the  series. 


CHAP.   Vlll.J  BINOMIAL  THEOREM.  257 

Tliis  scale  contains  two  terms,  and  the  series  is  called  a  re- 
curring  series  of  the  second  order.  In  general,  the  order  of  a 
recurring  series  is  denoted  by  the  number  of  terms  in  the  scale 
of  the  series. 

The  development  of  the  fraction 

a+  hx  +  cx^ 
a'  +  b'x -i- </x'^ -}- d' x^' 

gives  rise  to  a  recurring  series  of  the  third  order,  the  scale  of 
which  is, 

l-^"'     -^"''     -^*)5 

and,  in  general,  the  development  of 

o.  -\-  hx  -{-  cx'^  -^T  ,  ,  ,  ^x^^^ 
a'  +  b'x-\-</x^+   .  .  .  Fa;«  ' 

gives  a  recurring  series  of  the  n*^  order ^  the  scale  of  which  is 
I jx, -x^  .  .  r 7  a;") 

General  demonstration  of  the  Binomial  Theorem, 

202.  It  has  been  shown  (Art.  60),  that  any  expression  of  the 
form  z^  —  y"»,  is  exactly  divisible  by  z  —y^  when  m  is  a  po?i^ivf» 
whole  number,  giving, 

.  z'^  ^"^  fl^ 

£,         y 

The  number  of  terms  in  the  quotient  is  equal  to  m,  and  if 
we  suppose  z  =  y^  each  term  will  become  2;^-^ ;   hence, 

(^m  __  ym\ 
—  I       =   mz"^^. 
Z  —  y     Jy^t 

The  notation  employed  in  the  first  member,  simply  indicates 
what  the  quantity  within  the  parenthesis  becomes  when  we  make 
?/  =  z. 

We  now  propose  to  show  that  this  form  is  true  when  m  is 
fractional  and  when  it  is  negative. 

17 


258  ELEMENTS  OF  ALGEBRA.  [CHAP.  VIIL 

First,  suppose  m  fractional,  and  equal  to   — . 

JL  Z. 

Make  z^  =  v,     whence     z^  =zvp    and    z  =:vi\ 

JL  X 

and  yi=zu,     whence    yi=uP    and    y  =u9 , 


hence, 


JL         -P. 

^q    —  yq  yP  —  uP 


v^ 

— 

u^ 

V 

u 

V9 

— 

ui' 

z  — y         v^  —  w2 

V  —  u 
If  now,  we  suppose  y  =  2,  we  have  v  =  w,  and  since  p  and  q 
are  positive  whole  numbers,  we  have 

qv^-^       q  q 


V  —  u  fv^u 


Second,  suppose  m  negative,  and  either  entire  or  fractional. 
By   observing   that 

—  2-^  y-^  X  (f^  —  y^)  =  2r^  —  2/"*", 
we  have, 

«— w if'''^  Z^  —  "V^ 


2  — y  ^  —y 

If,  now,  we  make  the  supposition  that  y  =  2,  the  first  factoi 
of  the  second  member  reduces  to  -—  z-"^^,  and  the  second  fac- 
tor, from  the  principles  just  demonstrated,  reduces  to  m^'^- ; 
hence, 

\      z-y      ly^z 

We   conclude,  therefore,  that  the  form   is  general. 

203«  By   the   aid  of   the  principles   demonstrated  in   the  last 

article,   we    are    able    to    deduce    a  formula  for    the    develop. 

ment   of 

{x  +  a)*", 

w'hon  the  exponent  m  is  positive  or  negative,  entire  or  fractional 

Let  us   assume   the   equation, 

(l  +  z)^  =  F+  Qz  +  Ez^  +  Sz^  +  &c.  -     -     (1), 


CHAP.   VIII.J  BINOMIAL  THEOREAt.  259 

in  which,  P,  Q^  P,  &;c.,  are  independent  of  z,  and  depend  upon 
1  and  m  for  their  values.  It  is  required  to  find  such  values 
for  them  as  will  make  the  assumed  development  true  for  every 
possible  value  of  z. 

If,  in  equation  (1)  we   make   z  =z  0,  we   have 

Substituting   this  value  for  P,  equation  (1)  becomes, 

(l-{-z)^=l  +  Qz  +  Mz^  +  Sz^  +  &c.     -    -    -     (2). 
Equation  (2)  being  true  for  all  values  of  z^  let  us  make  z  =  y; 
whence, 

(1  -f  y)-  =  1  +  ^y  +  By^  +  Sy^  +  &c.     -    -    -     (3). 
Subtracting  equation  (3)  from  (2),  member  from  member,  and 
dividing  the  first  member  by  {1  +  z)  —  (1  +  y),  and  the  second 
member  by   its   equal   z  —  y,  we   have, 

{l  +  z)  —  {l+y)  z  —  y         z  —  y              z  —  y 

If,    now,    we    make  1  -{-  z  =  1  +  y,    whence    z  =  y,   the    first 

member    of   equation  (4),    from    previous    principles,    becomes 

m{l  -{-  z)^-^,  and  the  quotients  in  the  second  member  become 
respectively, 

\z~y/y=:z  \z  —  y/y^z  \z  —  y/y^z 

Substituting  these  results   in  equation   (4)  we  have, 
m{l+  zy-^  =  Q-\-2Ez  H-  3&2  +  4.Tz^  +  &c.     -    -    -     (5). 
Multiplying  both  members  of  equation  (5)  by  (1  +  0),  we  find, 


m  {l+z)'^=z  Q-\-  2E 

+    Q 


z  +  SS 
+  2E 


Z^+4:T 

+  SS 


23  +  &C.       -      .      .       (6). 


If  we  multiply  both  members  of  equation  (2)  by  w,  we  have 
m  {i  ■\-z)"'  =  m  +  mQz  +  mJRz^  +  mSz^  +  mTz^  +  &c.   -  -  -  (7). 

Tne  second  members  of  equations  (6)  and  (7)  are  equal  to 
each  other,  since  the  first  members  are  the  same;  hence,  we 
have   the   equation. 


m\-mQz-\-mBz'^+mSz'^+^Q,  =  Q+2R 

^    Q 


z+ZS 
4-2P 


2+47^ 
+3Sf 


^3+ &c-(8) 


260  ELEMENTS  OF  ALGEBRA.  [CHAP.   VIH 

This    equation   being  identical,  we  have,  (Art.  195), 


c  =  ^,   • 

.    or,      - 

-  «=y. 

2i^+^  =  me,. 

or,      . 

m{m  —  l) 
'    ^=      1.2' 

^S-h2E  =  mB, 

-     or,      - 

^       m(m-l)(m-2). 
•     ^=      1.2.3' 

iT+SS^mS, 

-     or. 

m(m- l)(w-2)(m-3) 
1.2.3.4* 

&c., 

&;c., 

&c. 

Substituting  these  values  in  equation  (2),  we   obtain 

(i+.).-^i+...+^^j^^-+"f7^]^"-'^^ 

_^m(^-lMm-2)Cm-3)^,_^^^_     .    -     (9). 


a 


If  now,  in  the  last  equation,  we  write  —  for  z,  and  then  mul- 
tiply both  members  bj  o;^,  we  shall  have, 

1  .  Mm— I)    „  ^  o  ,    m(m— l)(m— 2)    „  ^, 
(a:  -f  a)^=:a:^+??2ffa;^i+  — ^Ij — ^  a^.'c'"-^  ..j v_^ — ZA_^ ia^tJ^^a 

+  &c.  .  .  (10). 

Hence,  we  conclude,  since  this  formula  is  identical  with  that 
deduced  in  Art.  136,  that  the  form  of  the  development  of  (x+a)'^ 
will  be  the  same,  whether  m  is  positive  or  negative^  entire  or 
fractional 

It  is  plain  that  the  number  of  terms  of  the  development,  when 
m  is  either  fractional  or  negative,  will  be  infinite. 

Applications  of  the  Binomial  Formula. 

204.  If  in  the  formula  {x  +  «)"»  =: 

(a    .         w— 1    a2   ,         771—1    m  — 2   a^    ,  V 


CHAP.   VIII.]  BINOMIAL   THEOREM-.  26] 

we  make   m  =  — ,   it  becomes  (x  4-  a)n  or  \/  x  -\-  a  1= 

■       ^        1  ^1   1-2 

\nxn2x^n2  S        x^  J 

or,   reducing,  ^  x  +  a  = 

1/         I     a       1    71  -  1    a^        1    ?i  -  1    2w  -  1    1^3  \ 

The  fifth  term,  within  the  parenthesis,  can  be  found  by  mul- 
tiplying the  fourth  by  — and  by  — ,  then  changing  the  sign 

of  the  result,  and  so  on. 

205.  The  formula  just  deduced  may  be  used  to  find  an  approx- 
imate root  of  a  number.  Let  it  be  required  to  find,  by  means 
of  it,  the  cube  root  of  31. 

The  greatest  perfect  cube  in  31  is  27.  Let  x  =  27  and  a  =  4 : 
making  these  substitutions  in  the  formula,  and  putting  3  in  the 
place  of  n,  it   becomes 

14    _12^J^    llA     ^^ 


27  3     3  '729   '    3     3     9  *  19683 

115      2        256 


3  •  3  •  9  *  3  *  531441 


+  &C, 


•) 


or,   by   reducing, 

3  rwr_  Q   .   A  _  J^  ^     320     _      2560 

V  "^27       2187  "*■  531441       43046721  "^ 

Whence,  ^/ST  =  3  .  14138,  which,  as  we  shall  show  presently, 
is  exact  to  within  less  than  .00001. 

We  may,  in  like  manner,  treat  all  similar  cases :  hence,  for 
extracting  any  root,  approximatively,  by  the  binomial  formula, 
we  have   the   following 

RULE. 

^ind  the  perfect  power  of  the  degree  indicated ^  which  is  nearest 
to  the  given  number^  and  place  this  in  the  formula  for  x.  Sub- 
tract this  power  from  the  given  number,  and  substitute  this  differ- 
ence, which  will  often  be  negative,  in  the  formula  for  a.  Perform 
the  operations  indicated,  and  the  result  will    be  the  required   root 


262  ELEMENTS   OF   ALGEBiU..  LCHAP.    VUL 


EXAMPLES. 


1.         ■jy28=27^(l4-^\    =3.0366. 

1 

%        V^30"=  (32  -  2)*  =  32^A  -  3^)   =  1.9744. 

3.  ^"39"=  (32  +  Iff  =  32"^ /l  +  ^)   =  2.0807. 

1 

4.  \/T08'=  (128  -  20)^=  128^^1  -■  ^)  =  1.95204. 

206.  When  the  terms  of  a  series  go  on  decreasing  in  value, 
the  series  is  called  a  decreasing  series  ;  and  when  they  go  on 
increasing   in   value,  it  is   called   an   increasing    series. 

A  converging  series  is  one  in  which  the  greater  the  number 
of  terms  taken,  the  nearer  will  their  sum  approximate  to  a 
fixed  value,  which  is  the  true  sum  of  the  series.  When  the 
terms  of  a  decreasing  and  converging  series  are  alternately 
jiositive  and  negative^  as  in  the  firr;t  example  above,  we  can 
determine  the  degree  of  approximation  when  we  take  the  sum 
of  a  limited   number  of  terms  for   the   true   sum  of  the  series. 

For,  let  a  —  b-\-c  —  d-{-e—f-\-  .  .  .,  &c.,  be  a  decreasing 
series,  b,  c,  d,  ,  ,  ,  being  positive  quantities,  and  let  x  denote 
the  true  sum  of  this  series.  Then,  if  n  denote  the  number  of 
terms  taken,  the  value  of  x  will  be  found  between  the  sums 
of  n   and   n  -\-  1    terms. 

For,  take   any  two  consecutive    sums, 

a'-b'\-c~d+e  — /,      and      a  —  i  +  c  —  c?  4-«—/"f^. 

In  the  first,  the  terms  which  follow    — /,    are 
-i-  g  —  h,     -^  k  —  I  -{-   .  .; 
but,  since   the  series  is  decreasing,  the   terms    g  --  h,    k  -^  I  ,  , 
&c.,    are   positive ;    therefore,    in   order    to    obtain   the   complete 
value    of    iP,    a   positive    number    must    be    added    to   the    sura 
a  —  b  f  0  —  c?  -f-  e  — /.     Hence,  we   have 

a  —  b  +  c  —  d-\-e'-f<Cx. 


CHAP.   VIILJ  BINOMIAL  FORMULA.  263 

In  the  second  sum,  the  terms  which  follow  +  ^,  are  —  h 
-f  A?  —  Z  +  m  .  .  .  .  Now,  —  A  +  ^,  —  Z  +  m  .  .  &c.,  are 
negative ;  therefore,  in  order  to  obtain  the  sum  of  the  series, 
a  negative   quantity  must  be  added  to 

a —  b  +  c —  d  +  e —/+ g-, 

or,  in  other  words,  it  is  necessary  to  diminish  it.     Consequently, 

a  —  b  +  c  —  d-{'  e  — /+  9  >  x. 

Therefore,  x  is  comprehended  between  the  sums  of  the  first 
n   and    the  first   n  -\- \    terms. 

But  the  difference  between  these  two  sums  is  equal  to  g ;  and 
tfince  X  is  comprised  between  them,  g  must  be  greater  than 
the  difference  between  x  and  either  of  them ;  hence,  the  error 
committed  by  talcing  the  sum  of  n  terms,  a  —  b  +  c  —  d  +  e  — /, 
of  the  series^  for  the  sum  of  the  series  is  numerically  less  than 
the  following    term, 

207.  The  binomial   formula  serves   also   to   develop   algebraic 

expressions   into   series. 

EXAMPLES. 

1.  To  develop  the  expression ,     we  have, 

In  the  binomial  formula,  make  m=  —  1,  x  =  1,  and  a  =-•  —  4, 

and  it  becomes 

(1    _  ^)-l  =  1   _  1   .  (_  ^)  _  1   .  Z±pl    .  (_  ^)2 

-1-1    -1-2 

o^,    performing   the    operations    indicated,    we   find   for    the   de- 
velopment, 

--—  =  (1    -  ^)-l  =r   1  +  2  +  ^2  +  ^3  -}-  0*  +  (fee 

We  might  have  obtained  this  result,  by  applying  the  rule 
for   division. 


264  ELEMEISTTS  OF  ALGEBRA.  'CHAP.  Vni 

2.  Again  take   the   expression, 

(l4i)F     or    2(1-.)- 

Substituting   in    the    binomial   formula    --  3   for   m,    1    fcr   iP, 
and  —  z  for   a,   it  becomes, 

„    -3-1-3-2,       ,3       . 
-3.-2—. 3— .(-.)3-&c. 

Performing   the   indicated    operations    and    multiplying  by   2, 
we  find 

2 


(1-^)3 


2  (1  +  82  +  6s2  +  10^3  +  15^4  +  &c.). 


3.  To   develop    the    expression     3^ 2^  —z^     we  first  place   it 
under   the  form     3/2Jx(l  —-77]'     By  the   application  of  the 
'     binomial   formula,   we   find 

('-i)*--i(-i)+4-^;--(-i)'--- 


1 z z^ z^ 

6  36  648 


hence, 


'  ^^^^='y^(i-i^-^^^-6i8^'-'<^*'-) 


4.  Develop  the  expression 7^  =  (a  +  Z>)~2    into  a  series 

6.  Develop     into  a  series. 

T  -\-  X 

nyti  /vj3  /y4 

An8.  r  —  X  -\ H 5,  &c. 

CL       I     iC 

0.  Develop  the   square  root  of    -^ into  a  series. 


J^T^^    into   a  series. 
.         1         /,        2a;2    ,    5x*        40a;«     ,     \ 


7.  Develop  the  cube  root  of    t-^ rrr.     into   a  series. 


CHAP.   A^II.J  SUMMATION   OF  SERIES.  265 

Summation  of  Series. 

208.  The  Summation  of  a  Series^  is  the  operation  of  finding 
an  expression  for  the  sum  of  any  number  of  terms.  Many 
useful  series  may  be  summed  by  the  aid  of  two  auxiliary  series. 

Let  there  be  a  given  series,  whose  terms  may  be  derived  from 
the  expression  —. — ~- — r,  by  giving  to  p  a  fixed  value,  and  then 
attributing  suitable  values  to  q  and  n. 

Let  there  be  two  auxiliary  series  formed  from  the  expressions 

—  and  — ; — ,  so  that  the   values  of  »,   q,   and  n.   shall   be   the 
71  71  -{- p  ^  ' 

same  as  in  the  corresponding  terms  of  the  first  series. 

It   can    easily  be    shown   that  any  term  of  the   first   series   is 

equal   to   —  multiplied  by  the  excess  of  the  corresponding  terra 

in   the    second   series,  over   that  in   the   third. 
For,  if  we  take  the   expression 

L(± L\ 

p\n        n  +  pP 
and  perform  the  operations  indicated,  we  shall  get  the  expression, 

• ;     hence,  we  have 


n{n  -f-  p) ' 

p  ~^  p)     i^  W     n  +p/ 


n{. 
which  was   to   be  proved. 

It  follows,  therefore,  that  the  sum  of  any  number  of  terms  oj 
the  first  series^  is  equal  to  —  multiplied  by  the  excess  of  the  sum 

of  the  corresponding  terms   in   the    second   series,  over  that  of  the 
corresponding  terms  in  the  third  series. 

Whenever,   therefore,    we    can  -find   this  '  last    difference,  it  is 
always  possible  to  sum  the  given  series. 


^  =.  i 


266  ELEMENTS   OP  ALGEBRA.  LCHAP.    VIIL 


EXAMPLES. 

1.  Requirec,  the   sum  of  n  terms  of  the  series 

1.2^  2.3^  3.4  ^4.5^ 

Comparing  the  terms  of  this  series  with  the  expression 

9 

we  see  that  making   p  =  1,   2^  =  1,   and    w  =  1,  2,  3,  4,  dec,  in 
succession,  will  produce  the  given  series. 

The  two  corresponding  auxiliary  series,  to  n  terms,  are 

'+i+i+i+ ^. 

2         3         4  n        n  -{-  1 

The  difference  between  the   sums  of  n  terms  of  the  first  and 
second   auxiliary  series   is 

1 — — ,  or,  if  we  denote  the  sum 

n  -\-  1 

of  n  terms  of  the  given  series   by  S,  we   have, 

n  +  1 
If  the  number  Df  terms  is  infinite   n  =z  co   and 

5  =  1. 

2.  Required  the  sum  of  n  terms  of  the  series 

O  +  O  +  577  +  779  +  970  +  *"•' 
If  we   compare   the   terms  of  this   series  with   the  expression 

n{n  +  pY 
we   see   that  /^  ^  2,   5'  —  1,   and  n  =  l,    3,    5,    7,    &c.,    in   suc- 
cession. 


CHAP.   VIII.J  SUMMATION  OF  SERIES.  267 

The  two  auxiliary  series,  to  n  terms,  are, 

i+i.  +  i.  +  i  + +    ' 


g-r^-r^-r -^  2n  ^  V 

1.1      .     1      .  .1.1 


hence,   as   before. 

If  w  =  00,    we   find  S'=  — . 

3.  Eequired   the   sum  of  n  terms  of  the   series 

—  4-  —  +  —  +  —  +  &c. 
1.4^  2.5^  3.6^  4.7  ^ 

Here  P  =  ^^     S'  =  1>     n  =  1,  2,  3,  4,   &c. 

The   two   auxiliary  series,   to   n   terms,   are, 

1+1  +1+_J_+_1_+_1.. 

hence,     ^^  ==  i  (l  + -1  +  1  - -J-^  _ -1^  _ -1^). 

If  n=cx>,  i>  =—. 

4.  Required  the  sum  of  the  series 

1.5  ^5.9^   9.13^  13.  17  ^17.21  ^ 

5.  Eind   the   sum  of  n  terms  of  the   series, 

__2 L^_l ^    .   _J ^ 

3.5      5.7  "^7.9       9.11  "^11.13       ^^-  •  •  • 

Herejp  =  2,   q  =2,  ~  3,  +4,  -5,  +  6,  &c 

»=  3,  5,  7,  9,  11,  &c. 


268  ELEMENTS   OF  ALGEBRAi  LCHAP.   VHI. 

The   two    auxiliary  series   are, 

2         3,4         5    ,  ^  +  1 


3         5'7         9' ^2/i  +  l 


5         7    '    9  •        2^  +  1       2/*  -f-  3 ' 

If   n    is   cve/i,    the  upper   sign   is  used,    and    the   quantity   io 
the  last  parenthesis    becomes    +  1,  in   which   case 

^^1/2     ^  +  i\     1     JL/_JL  ,  Mj^\ 

2  V3  ^^  +  3/        2  -  2  V      3  "^  ^Ai  +  3/' 
If  ?i   is   odd^  the  lower  sign  is   used,  and   the  quantity  in   tk 
last  parenthesis   becomes  0,  in  which   case 

'2        n  +  \\. 


S: 


2  V^ 


,3       2/1  +  3/' 
If  in   either  formula  we   make 

2 

71  +  1         1+  ^      ,  1  .      cr        1 

^  =  ^'2;rF3  = 3"    ^T^^^     T'     "^^^     '^  =  12- 

6.  Find   the   sum   of  n   terms   of  the   series, 

J Lj,  J L  A. 

1.3      2.4'^3.5      4.6' 
Here,  JP  =  2,  g'  =  1,  —  1,  +1,  —  1,+  1,  —  1,  &c. 

w  =  1,  2,  3,  4,  (fee. 
TTie  two   auxiliary   series   are, 

^       2"^3        4"^  5        Q'^  '  '  '  '   ^  n 

*"3        4'^5        6"^--^-^^-^+l'^n'"-f"i 
whence,  5  =  i  (i  ^  ^  ±. -1^). 

If   n  =  00,    we  find    S  =  ■^. 


CHAP.   VIII.]  METHOD   BY  DIFFERENCES.  269 

Of  the  Method  ly  Differences, 

209.  Let  a,  6,  c,  c?  .  .  .  .  &c.,  represent  the  successive  terms 
of  a  series  formed  according  to  any  fixed  law ;  then  if  each 
term  be  subtracted  from  the  succeeding  one,  the  several  re 
mainders  will  form  a  new  series  called  the  first  order  of  dif- 
ferences. If  we  subtract  each  term  of  this  series  from  the 
succeeding  one,  we  shall  form  another  series  called  the  second 
order  of  differences^  and  so  on,  as  exhibited  in  the  annexed 
table, 
a,     6,  c,  c?,  e, 

b — a,       c—b,  d—c,  e  — c?,  &;c.,        1st. 

c— 26+a,  d—2c  +  6,  e—2d+  c,&c.,    2d. 

d—Sc+Sb—a,  e—Sd+Sc  —  b,  &c.,    3d. 

e—Ad+Qc^Ab  +  a,  (fee,  4th. 

if,  now,  we  designate  the  first  terms  of  the  first,  secorfd. 
third,  (fee.  orders  of  difierences,  by  c?i,  c?2)  ^s?  ^4?  <^c.,  we  shall 
have, 

d^  =z  b—   a,  whence  b  =  a-\-    d^, 

d^  z=  c  —  2b  -{-  a,  whence  c  =  a-\-2di+    cfg, 

c?3  —  0?  —  3c  +  3&  —  a,  whence  c?  r=  a  4-  3c?i  +  ^d^  +    (^zi 

c?4  ==  e  —  4cZ  -f-  6c  —  4i  +  «j  whence  c  =  a  +  4(fi  +  6d^  +  4d^  +  d^, 
&c.  &c.  &c.  &c. 

And  if  we  designate  the  term  of  the  series  which  has  n 
terms  befjre  it,  by  T,  we  shall  find,  by  a  continuation  of 
the   above  process, 

This  formula  enables  us  to  find  the  (n-f-1)'^  term  of  a 
series  when  we  know  the  first  terms  of  the  successive  orders 
jf  difierences. 


270  ELEMENTS   OF  ALGEBRA.  [CHAP.   YIII. 

210.  To  find  an  expression  for  the  sum  of  n  terms  of  the 
series  a,   b,   c,  &;c.,  let   us   take   the   series 

0,  a,  a-\-  b,     a  -{-  b  +  c,     a  +  b  +  c  +  d,  &c.     ....    (2) 
The   first   order  of  differences   is   evidently 

a,  b,  c,  d^   .    .    .    .    .    .    &c. •    (3) 

Now,  it  is  obvious  that  the  sum  of  n  terms  of  the  series  (3), 
Is   equal   to   the   (ti  +  I)*^   term  of  the   series   (2). 

But  the  first  term  of  the  first  order  of  differences  in  series  (2) 
is  a;  the  first  term  of  the  second  order  of  differences  is  the 
same  as  di  in  equation  (1).  The  first  term  of  the  third  order 
of  differences  is  equal  to  d^,  and  so  on. 

Hence,  making  these  changes  in  formula  (1),  and  denoting  the 
sum  of  n  terms  bj  S,  we  have, 

,^^^a+.____^^+        17273        "^'^  1.2.3.4  "^^ 

-r  &c.     .    -    -    -     (4). 

When  all  of  the  terms  of  any  order  of  differences  become 
equal,  the  terms  of  all  succeeding  orders  of  differences  are  0, 
and  formulas  (1)  and  (4)  give  exact  res^ilts.  When  there  are 
no  orders  of  differences,  whose  terms  become  equal,  then  for- 
mulas do  not  give  exact  results,  but  approximations  more  or  less 
exact  according  to  the  number  of  terms  used. 

EXAMPLES. 

1.  Find  the  sum  of  n  terms  of  the  series  1.2,  2.3,  3.4, 
i .  5,   (fee. 

Series,  1.2,      2.3,       3.4,       4.5.       5 . 6,  &c. 

1st  order  of  differences,       4,  G,  8,  10,  &c. 

2d  order  of  differences,  2,  2,  2,  &c. 

3d  order  of  differences,  0,  0. 

Hence,  we  have,  a  =  2,  c?,  ~  4,  d^=z  2,  d^,  d^,  &c.,  equal 
10  0. 


CHAP.   VIII..  METHOD  BY  DIFFERENCES.  271 

Substituting   these   values   for   c,    c?,,    c?^,    &;c.,  in   formula  (4), 
we  find, 

a      r»      .   n(n^l)        ^    .    n(n  —  l)(n—2)       _ 

waence,  /S  =  -^ ^^^ ^. 

o 

2.  Find    the   sum  of  n  terms  of  the   series    1.2.3,     2.3,4^^ 
S.4.5,     4.5.6,   &c. 

1st  order  of  differences,  18,        36,        60,        90,       126,  &:c. 

2d  order  of  differences,  18,       24,        30,        36,  &c. 

3d  order  of  differences,  6,          6,         6,  &;c. 

4th  order  of  differences,  0,         0.    &c. 

We  find    a  =  6,     d,  =  18,     d^  =  18,     d^  =6,     d^  =  0,  &;c. 

Substituting  in  equation  (4),  and  reducing,  we  find, 

^      n{n  +  l){n  +  2){n  +  S) 
S  = . 

3.  Find  the  sum  of  n  terms  of  the  series  1,  1+2,  1+2+3, 

1  +  2  +  3  +  4,   &c. 

Series,  1,         3,         6,         10,         15,         21. 

1st  order  of  differences,       2,         3,         4,  5,  6. 

2d  order  of  differences,  1,         1,  1,  1. 

3d  order  of  differences,  0,         0,  0, 

a  =  1,     c?i  =  2,     c?2  =  1,     c?3  =  0,     c/4  =  0,  (fee. ; 

hence      S -n  I   ^(^"^^   2  I  ^(^^l)  (^-^)  _  ^^  +  ^^^  +  2n. 
hence,     ^ -n  +     ^^     .2+         ^^^         _         ^^^        , 

,     .                   ^      w(w  +  1)  (n  +  2) 
or,   reducmg,  S  =  -^^ —     ^  ^  -, 

4.  Find  the  sum  of  n  terms  of  the  series  1^,  2^,  3^,  4^,  5^,  &c. 
We  find,  a  =  1,  cfj  =  3,  t?.,  =  2,  Jj  =  0,  d^=:  0,  &c.,  &c. 
Substituting  these  values  in  formula  (4),  and  reducing,  we  find, 

n{n  +  l){2n  +  1) 

^= — rT2~3" — • 


272 


^ELEMENTS   OF   ALGEBRA. 


[CHAP.   VIII. 


5.  Find   the   sum   of  n   terms   of  the   series, 

1 .  (m  +  1),     2  {in  +  2),     3  {m  +  3),     4  (m  -f  4),  &c. 
We  find,     a  =  m -\- 1,     c?i  =  m  +  3,     d^  —  2^     d^=:0,  &c. ; 

whence,  ^^.(^  +  1)  4!i4^(- +  ^)  +  ^^^^^ 


>S'  = 


1  .  2     '      '     '   '        1.2 
71 .  (/i  +  l).(l+2/z+3m) 


1 


2 


ty  Piling  Balls, 

The  last  three  formulas  deduced,  are  of  practical  appli- 
cation in  determining  the  number  of  balls  in  different  shaped 
piles. 

First^  in   the  Triangular   Pile. 

211.  A  triangular  pile  is  formed  of  succces- 
sive  triangular  layers,  such  that  the  number 
of  shot  in  each  side  of  the  layers,  decreases 
continuously  by  1  to  the  single  shot  at  the 
top.  The  number  of  balls  in  a  complete  tri- 
angular pile  is  evidently  equal  to  the  sum 
of  the  series  1,  1  +  2,  1  +  2  +  3,  1  +  2  +  3 
+  4,  &c.  to  1  +  2  +  .  .  .  +  7i,  fi  denoting  the  number  of  balls 
on   one   side   of  the   base. 

But  from  example  3d,  last  article,  we  find  the  sum  of  n 
terms   of  the   series, 

n{n  +  \){n  +  2) 


S  = 


1.2.3 

Second^  in   the   Square   Pile, 


a)- 


21 2.  The  square  pile  is  formed, 
a3  shown  in  the  figure.  The  num- 
ber of  balls  in  the  top  layer  is  1  ; 
the  number  in  the  second  layer  is 
denoted  by  2^ ;  in  the  next,  by  3^, 
and  so  on.  Hence,  the  number  of 
balls  in  a  pile  of  n  layers,  is  equal 
to  the  sum  of  the  series,  P,  2^  3^, 


/0» 


(W)Ji^(^ 


CHAP.   VIII.]  PILING  BALLS.  273 

^c,  n^,  which  we   see,  from   example  4th  of  the  last  article,  is- 

^=^       1.2.3  -     •     "     ^^^• 

Third,  in   the   Oblong  Pile. 


213.  The  complete  oblong  pile  has  (7/1+ 1)  balls  in  the 
upper  layer,  2 .  (m  +  2)  in  the  next  layer,  3  (m  +  3)  in  the 
third,  and  so  on :  hence,  the  number  of  balls  in  the  complete 
pile,  is  given  by  the  formula  deduced  in  example  5th  of  the 
preceding   article, 

n.(/i  +  l).(l+2/i+3m) 
^= 1.2.3 •     ■     •     (^)- 

21 4#  If  any  of  these  piles  is  incomplete,  compute  the  nuin- 
ber  of  balls  that  it  would  contain  if  complete,  and  the  number 
that  would  be  required  to  complete  it  ;  the  excess  of  the  for 
mcr  over   the   latter,  will   be   the   number   of  balls   in   the   pile. 

The  formulas  (1),  (2)  and  (3)   may  be  written, 

triangular,         S  =  j ,  ""  ^"\^  ^\n  +  I  +  I) (1)  ; 

square,  S  =  ~  .^^^^^^  {n  +  n  +  I) (2); 

rectangular,      ^=j'  n{n+l)  ^^^_^^^_^^^_^^^^^^_j_  j^\  _  ^3^^ 

n(n  -\-  I)  ,     .  ,  « ,    n     .      1 

Now,  smce  • — ^—^ is  the  number  of  balls  m  the  in- 

/w 

RTigiilar  face  of  each  pile,  and  the  next  factor,  the  number  of  balls 

in    I  he   longest,  line    of  the    base,  plus   the   number   in    the   side 

of  the   base  opposite,   plus   the   parallel   top   row,  we    have   tht 

following 

18 


274  ELEMENTS   OF  ALGEBRA.  •  [CHAP,   VID. 

RULE. 

Add  to  the  number  of  balls  in  the  longest  line  of  the  base  the 
number  in  the  parallel  side  opposite^  and  also  the  number  in  the 
top  parallel  row ;  then  multiply  this  sum  by  one-third  the  number 
in  the  triangular  face ;  the  product  will  be  the  number  of  balls  ia 
the  pile, 

EXAMPLES. 

^  1.  How  many  balls  in   a   triangular   pile  of  15   courses'? 

^'  Ans,  680. 

2.  How  many  balls  in  a  square  pile  of  14  courses  1  and  how 
many  will   remain   after   5   courses   are   removed  1 

Ans,  1015  and  960. 

3.  In  an  oblong  pile,  the  length  and  breadth  at  bottom  are 
respectively   60   and   30 :   how   many   balls   does   it    contain  1 

Ans,  23405. 

4.  In  an  incomplete  oblong  pile,  the  length  and  breadth 
at  bottom  are  respectively  46  and  20,  and  the  length  and 
breadth  at   top   35   and   9 :   how  many  balls   does   it   contain  ? 

Ans,  7190. 

^  5.  How  many  balls  in  an  incomplete  triangular  pile,  the  num  . 
ber  of  balls  in  each  side  of  the  lower  course  being  20,  and 
m  each   side  of  the  upper,  10? 

6.  How  many  balls  in  an  incomplete  square  pile,  the  number 
in  each  side  of  the  lower  course  being  15,  and  in  each  side 
of  the   upper  course  6  ? 

7.  How  many  balls  in  an  incomplete  oblong  pile,  the  num- 
bers in  the  lower  courses  being  92  and  40 ;  and  the  numbers 
hi  the    jorresponding  top  courses  being  70  and  18  "3^ 


CHAPIDR  JX. 

CONTINUED   FRACTIONS — EXPONENTIAL   QUANTITIES LOGARirHMS,    AND 

FORMULAS   FOR  INTEREST. 

215.  Every  expression  of  the  form 

3  1  1 


a+l  a+r  a+V 


b  b+\  b+\ 


c  c-r  1 

in  which  a,  b;  c,  d,  &c.,  are  positive  whole  numbers,  is  called  a 
continued  fraction :   hence, 

A  CONTINUED  FRACTION  kus  1  for  its  numerator^  and  for  its  de- 
nomiiiator^  a  whole  number  plus  a  fraction^  which  has  1  for  its 
numerator  and  for  its  denominator  a  whole  number  plus  a  fra^ 
tion,  and  so  on, 

216.  The  resolution  of  equations  of  the  form 

a*  =  6, 

gives  rise  to  continued  fractions. 

Suppose,  for  example,   a  =  8^   6  =  32.     We  then  have 

s'  =  32, 
it    which  a;  >  1    and  less  than  2.     Make 

*  c=  1  +  1, 

y 


^ 


276  ELEMENTS   OF    ALGEBRA.       .  'CHAP.    IX. 

ill   which  2/  >  1,  and  the  proposed  equation  becomes 

32  =  8      ^  =8  X  SJ';     whence, 

i  y 

8^  =  4,     and  consequently,     8  =  4. 

It  is  plain,  that  the  value  of  y  lies  between  1  and  2.     Suppose 

1+i        i 

and  we  have,  8  =  4      *=4x4*; 

1 

Itence,  4*  =  2,     and    4  =  2,     or    2=2. 

But,  y  =  l  +  i  =  l  +  ^=|; 

and  .  =  l+.i=l  +  _L_=l  +  |=|; 

1  +  2 
and  this  value  will   satisfy  the  proposed   equation. 

For,  8^  =  3/8^  =  ^/(23)^  =  3/(2^  =  2   =32. 

21 7»  If  we  apply  a  similar  process  to  the   equation 
(10)*  =  200, 
we  shall  f'nd 

:r  =  2+~;      y  =  3  +  i-:      ^2  =  3  +  -. 

y  z  '  u 

Since  200  is  not  an  exact  power,  x  cannot  be  exactly  ex- 
pressed  either  by  a  whole  number  or  a  fraction:  hence,  the 
▼alue  of  X  will  be  incommensurable  with  1,  and  the  continued 
firaction  will  not  terminate,  but  will  be  of  the  form 

«  =  2  +  l=2  +  ^ =-  =  2+  ^ 


y  3  +  1  3+^ 


u  +  &o. 


CHAP.   IX.  1  CONTINUED  FRACTIONS.  277 

218.  Vulgar  fractions  may  also  be  placed  under   the  form  of 

continued  fractions, 

65 
Let  us  take,  for  example,  the  fraction  — -,   and  divide  botb 

its  terms  by  the  numerator   65,  the  value  of  the  fraction  will 
not  be  cnanged,  and  we  shall  have 

65   _     1 

149  ""  j49' 

65 

^     .         1       ....           65              1  / 

or   efiectmg  the  division,    —— ^  = —. 

19 

Now,  if  we  neglect  the  fractional  part,    -7-,    of  the  denomina 

00 

toi,  we  shall  obtain   —    for  an  approximate  value  of  the  given 

fraction.     But  this  value  will  be  too   large,  since  the  denomina- 
tor  used   is   too    small. 

If,  on   the  contrary,   instead  of  neglecting  the  part    — ,     we 

were   to   replace   it  by  1,  the  approximate  value  would  be    — , 

which   would  be  too    small,  since    the    denominator    3    is    too 
large.     Hence, 

1  ^     65  ,      1    ^    65 

2->Ti9  ""^     T<.-l49' 

therefore  the  value  of  the  fraction  is  comprised  between  —  and  -^. 

If  we   wish  a  nearer  approximation,  it  is   only   necessary  to 

operate  on  the  fraction  —   as  we  did  on  the  given  fraction  -rja^ 

and   we  obtain, 

19  _       1 

^  +  19' 


hence, 


65       1 


'''      2  +  i 


^-^w 


278  ELEMENTS   OF   ALGEBRA.*  'CHAP.   IX. 

o 

If,  now,  we  neglect  :ne  part  — r,  the  denominator  3  will  be  less 

J.  t/ 

than  the  true  denominator,  and   — -    will  be  larger  than  the  num 

o 

ber  which  ought  to  be  added  to  2;   hence,  1  divided  by  2  4*  — 

o 

will   be  less   than  the  true  value  of  the  fraction ;   that  is,  if  we 

stop   after   the   first  reduction    and    omit    the  last  fraction,   the 

result  will  be  too  great ;  if  at  the  second,  it  will  be  too  small,  &c. ; 

and,  generally. 

If  we  stop  at  an  odd  reduction^  and  neglect  the  fractional  part 

that  comes  after,  the   result  will  he  too  great;   hut  if  we  stop  at 

an  even  reduction,   and  neglect  the  fractional  part  that  follows,  the 

result  will  he   too   small, 

219.  The  separate  fractions   — ,     — -,     — ,  &c.,  which  make  up 

a  conUnued  fraction,  are   called   integral  fractions. 
The   fractions, 

1  1 1 


a  -\ — r-         a  + 


c 

are  caJled  ap-proximating  fractions,  because  each  gives,  in  succes- 
sion, a  nearer  approximation  to  the  true  value  of  the  fraction : 
hfmce, 

An  approximating  fraction  is  the  result  obtained  hy  stopping 
at  any   integral  fraction,  and  neglecting   all   that  come   after. 

If  we  stop  at  the  first  integral  fraction,  the  resulting  approxi- 
mating fraction  is  said  to  be  of  the  first  order ;  if  at  the  second 
integral  fraction,  the  resulting  approximating  fraction  is  of  the 
*?econd  order,  and  so  on. 

When  there  is  a  finite  number  of  integral  fractions,  we  shall 
get  the  true  value  of  the  expression  by  considering  them  all : 
when  their  number  is  infinite,  only  an  approximate  value  can  be 
found. 


CHiF.   IX.]  CONTINUED  FRACTIONS.  279 

220.  We  will  now  explain  the  manner  in  which  any  approxi- 
mating  fraction  may  be  found  from  those  which  precede  it. 

(D-    -'- 

^  a 

P)..-!- 

(3)-- 


-I 


a  + 


1 

a 

1st  app. 

fraction. 

b 
ah -{-I 

2d   app. 

fraction. 

bc+l 
(ah-\-\)c 

+  a 

3d   app. 

fractior.. 

.+  -1 


c 

By  examining  the  third  approximating  fraction,  we  see  that 
its  numerator  is  formed  by  multiplying  the  numerator  of  the 
preceding  approximating  fruction  by  the  denominator  of  the 
third  integral  fraction,  and  adding  to  the  product  the  numerator 
of  the  first  approximating  fraction :  and  that  the  denominator 
is  formed  by  multiplying  the  denominator  of  the  preceding 
approximating  fraction  by  the  denominator  of  the  third  integral 
fraction,  and  adding  to  the  product  the  denominator  of  the 
first    approximating   fraction. 

Let  us  now  assume  that  the  {n  —  1)'^  approximating  fraction 
is  formed  from  the  two  preceding  approximating  fractions  by  the 

same  law,  and  let  — ,     — ,    and    — ,  designate,  respectively,  the 

(71  ~  3),    (/I  —  2),  and   {n  -—  1),   approximating   fractions. 

Then,  if  m  denote  the  denominator  of  the  (n  —  1)*^  integral 
fraction,  we  shall  have  from  the  assumed  law  of  formation. 


R  ~  Q'm  -\-P" 


(1). 


1  S 

Let  us  now  consider  another  integral  fraction  — ,  and  suppose  — 

n  o 

to  represent   the   n^^   approximating   fraction.      It    is   plain   that 

S  R 

we   shall   obtain   the    value    of  — ,  from  that  of  — ,    by   simply 

>o  R 

changing  —  into — ,  or,  'oy  substituting  m^ —     for    m,    in 

m  \ 

n 

oquation    (1) ; 


280  ELEMENTS  OF  ALGEBRA.'  [CHAP.  IX 

wlience,     ^,  _       •     ^  j_x       ^^  "  ( Q^m  +p.>  +  §' "  ij'„+  q" 

Hence,  if  the  law  assumed  fcr  the  formation  of  the  (yi — 1)'^  ap* 
proximating  fraction  is  true,  the  same  law  is  true  for  the  forma- 
tion  of  the  n*^  approximating  fraction.  Cut  we  have  shown 
that  the  law  is  true  for  the  formation  of  the  third ;  hence,  il 
must  be  true  for  the.  formation  of  the  fourth;  being  true  for 
thp  fourth,  it  is  true  for  the  fifth,  and  so  on ;  rience,  it  is  gen 
eral.     Therefore, 

The  numerator  of  the  n*^  approximating  fraction  is  formed  by 
multiplying  the  numerator  of  the  preceding  fraction  by  the  denom 
inator  of  the  n*^  integral  fraction^  and  adding  to  the  product  the 
numerator  of  the  (n  —  2)*^  approximating  fraction  ;  and  the  denom- 
inator is  formed  according  to  the  same  law ^  from  the  two  preceding 
denominators, 

221.  If  we   take   the   difference   between   the   first   and  second 
approximating  fractions,  we  find, 

1    ^        b         __ab-\-\  ~ab  _       -f  1 
~a   ""  ab  -^  1  ~    a{ab  +  1)    ~  a(ab  +  1)  ' 

and   the   diflference   between  the   second  and  third  is, 

b  bc-\-l  - 1 


ab  -f  1        {ab  -f  l)c  +  a        (ah  -\-  1)  {{ab  -f  l)c  -[-  «]* 
In   both  these   cases   we   see   that   the   difference   between  two 
consecutive   approximating   fractions   is   numerically    equal   to    1, 
divided  by  the  product  of  the  denominators  of  the  two  fractions 
To   show  that  this   law   is  general,  let 

P^         Q_        ^ 

P'  q'         R' 

be  any   three   consecutive   approximating   fractions.     Then 

p      Q     pq^-p^Q 


aud 


Q^  _^_  P'q -Pq' 


CHAP.    IX.J  CONTINL'ED   FRACTIONS.  281 

But         JS  =  §m  -f  P,     and     R  =  Q'm  -\-  F'  (Art.  220). 
Substituting   these   values   In   the   .ast  equation,  we   have, 
Q__R__  {Q'm-^P')Q-{Qm  +  P)Q\ 

or,  reducing, 

Q        R        P^Q^PQ^ 
Q'       R'~       RQ'      • 

Now,  if  {PQ'  —  P'Q)  is  equal  to  dt  1,  then  {P'Q  —  PQ')  must 
be   equal   to    q=  1 ;   that  is, 

If  the  difference  between  the  {n  —  2)  and  the  {n  —  1)  fractions^ 
is  formed  by  the  assumed  law,  then  the  difference  between  the 
{n  —  1)*^  and  the  n*^  fractions  must  be  formed  by  the  same  law. 

But  we  have  shown  that  the  law  holds  true  for  the  difference 
between  the  second  and  third  fractions ;  hence,  it  must  be  true  fo^ 
the  difference  between  the  third  and  fourth;  being  true  for  the 
difference  between  the  third  and  fourth,  it  must  be  true  for  the 
difference  between  the  fourth  and  fifth,  and  so  on ;  hence,  it  is 
general :    that  is, 

The  difference  between  any  two  consecutive  approximating  frac- 
tions^ is  equal  to  zb  1,  divided  by  the  product  of  their  denoni 
inators. 

When  an  approximating  fraction  of  an  even  order  \j  taken 
from  one  of  an  odd  order,  the  upper  sign  is  used :  when  one 
of  an  odd  order  is  taken  from  one  of  an  even  order,  the- 
lower  sign   is  used. 

This  ought  to  be  the  case,  since  we  have  shown  that  ever^ 
approximating  fraction  of  an  odd  order  is  greater  than  the  true 
value  of  the  continued  fraction,  whilst  every  one  of  an  even 
order  is  less. 

222.  It  has  already  been  shown  (Art.  218),  that  each  of  the 
approximating  fractions  of  an  odd  order,  exceeds  the  true  value 
of  the  continued  fraction ;  while  each  one  of  an  even  order 
is  less  than  it.  Hence,  the  difference  between  any  two  con- 
secutive  approximating  fractions  U  greater    than   the    difference 


282  ELEMENTS   OF  ALGEBRA.    ^  [CHAP.   IX. 

between  either  of  them  and  the  true  value  of  the  continued 
fraction.  Therefore,  stopping  at  the  n*^  approximating  fraction, 
the  result  will  be  the  true  value  of  the  fraction,  to  within  less 
than  1  divided  by  the  denominator  of  that  fraction,  multiplied 
by  the  denominator  of  the  approximating  fraction  which  follows. 

Thus,  if  Q^  and  H^  are  the  denominators  of  consecutive  ap- 
proximating fractions,  and  we  stop  at  the  fraction  whose  de- 
nominator is  Q',  the  result  will  be  true  to  within  less  than  • 

Q  it 

But,  since  a,  b,  c,  cZ,  &c.,  are  entire  numbers,  the  denominatoi  JH' 
will  be  greater  than    @^,   and  we  shall  have 

hence,  if  the  result  be  true  to  wdthin  less  than  ,     it    will 

Q  it 

certainly  be  true  to  within   less   than   the   larger  quantity 
-^ ;     that  is. 

The  approximate  result  which  is  obtained,  is  true  to  within 
less  than  1  divided  by  the  square  of  the  denominator  of  the  last 
ai)proximati7ig  fraction    that   is   employed, 

829 
223  •  If  we  take  the  fraction   7——,    we  have, 

347 

347  "^  „  .       1 


1  + 


1+      1 


^-w 


Hero,  we  have  in  the  quotient  the  whole  number  2,  w^hich 
may  either  be  set  aside,  and  added  to  the  fractional  part  after 
its ,  value  shall  have  been  found,  or  we  may  place  1  under  ifc 
%r  a  denominator,  and   t^'cat  it  as   an  approximating  fraction. 


CHAP.   IX.J  EXPONENTIAL   QUANTITIES.  283 

Solution  of  the  Equation  a*  =  b. 

224.  Ai    equation  of  the  form, 

a*  =  6, 
.s   called    an    exponential   equation.      The   object   in   solving   this 
equation  is,  to   find   the   exponent  of  the   power   to  which   it   is 
necessary   to    raise    a   given    number    a,   in    order    to    produce 
another  given  number  5. 

225.  Suppose  it  were  required,  to   solve  the   equation, 

2^  =  64. 
By  raising  2  to  its  different  powers,  we  find   that 

6 

2  =  64 ;      hence,      a;  =  6 
will   satisfy  the  equation. 

Again,  let  there  be   the   equation, 

3  =  243,     in  which    ic  =  5. 

Now,  so  long  as  the  second  member  5  is  a  perfect  power  of 
the  given  number  a,  the  value  of  x  may  be  obtained  by  trial, 
by  raising  a  to  its  successive  powers,  commencing  at  the  first, 
hr  the   exponent  of  the  power  will  be  the  value  of  x, 

226.  Suppose  it  were  required  to  solve  the   equation, 

z 

2=6. 
By  making    a;  =  2,    and    x  =  B,    we  find, 

2  3 

2=4     and     2=8; 
from  which  we  perceive   that  the  value   of  x   is   comprised   he- 
tweei  2   and  3. 

Make,  then,  x  =2  -i — ^,     in  which    a/  >  1. 

Substituting  this  value  in  the  given   equation,  it  becomes, 

a+i  _L 

2      =='  r=  6,     or     22  X  2^^  =  6  ;     hence, 

2^-  ^  -  ^  . 
^     -4  -"2 


284  ELEMENTS   OF   ALC^BRA.  [CHAP.   IX 

and  by  changiiig   the   order   of  the   members,  and    raising   both 
to   the   xf  power, 

To  determine  x\  make   a/  successively  equal  to  1  ard  2;    we 
find, 

therefore,  y^  is  comprised  between  1  and  2. 
Make,  x^  z=z\  -\-  — ,     in  which  a;^'  >  1. 

By  substituting  this  value  in  the  equation  j  — ■  j    =2, 
we  find,  (1)^'"-  =  2;     hence,     \  X   (|-)^  =  2, 

and   consequently,    \^  \    i~|     = -h"* 

mi  4  3 

The   supposition,         a;^^  =  1,     gives      7r<7r; 

„      ^        .  16       3 

and  a;^^  =  2,     gives     y>-2> 

therefore,  x^^  is  comprised  between   1    and  2. 

Let  xf^  z=z\'\ — ■ ;     then, 

xf^^ 

/4V+^/        3       ,  4        /4W      3 

(-)  =~;     hence,     ^  X  (3)      =^, 

whence,  (-j      = --. 

If  we   make   xf^^  =  2,  we  have 

and  if  we  make  x^^^  —  3,  we  have 


/9y_729      4  ^ 


"512"^  3 


CHAP.   IX.  J                 EXPONENTIAL   QUANTITIES.  285 

therefore,  x^^^   is   comprised   between   2   and   3. 
Make  xf'^  =  2  H ,   and  we  have 

•     /IV""^^---     hence      ^(l\^v-l. 
\8/  ~  3  '     ^^^^'     64V8r   ~  3  ' 

and  consequently,  (243)''    ~  T* 

Operating  upon  this  exponential  equation  in  the  same  manner 
as  upon  the  preceding  equations,  we  shall  find  two  entire  num 
bers,  2  and  3,  between  which  x^"^  will  be  comprised. 

Making 

X    can  be  determined  in  the   same   manner  as  rc^^,  and  so  on. 
Making   the   necessary   substitutions   in   the   equations 

.  =  2+1,    ^  =  1+^,    -"=1  +  ^.    ."'=2  +  -i^...., 

we  obtain  the   value  of  x  under  the  form  of  a  whole  number, 
plus  a  continued  fraction. 

1 


a;  =  2  + 


'+r— 1 


2+  ^ 

x" 


V  > 

hence,  we  find  the  first  three  approximating  fractions  to  be 

JL      JL       A 

1'  2'  5' 

\  and   the   fourth  is   equal   to 

3x2  +  1        7 


(Art.  220), 

tne   true  value  of  the   frac 
less   than 


5x2  +  2       12 
which   is   tne   true   value  of  the   fractional   part   of  x   to   wilhir 


W'    °^     lli  (^'•t-  222). 


286  ELEMENTS   OF  ALGEBRA.  [CHAP.   IX. 

Therefore, 

7        31  1 

a;  =  2+— =—  =  2.58333  +     to  within  less  than    -j-, 

and  if  a  greater  degree  of  exactness   is   required,  we  must  take 
a  greater  number  of  integral  fractions. 

EXAMPLES. 

3   =  15         -    -    xz=         2.46    to  within  less  than    0.01. 


(10)    =3     -    -    .    a;=         0.477  "  "  0.001. 

_2 
3 


5*  =  -^    -    .    .    x=   -  0.25  "  "  0.01, 


0/  Logarithms, 

227.  If  we  suppose  a  to  preserve  a  constant  value  in  the 
equation 

whilst  iV  is  made,  in  succession,  equal  to  every  possible  num- 
ber, it  is  plain  that  x  will  undergo  changes  corresponding  to 
those  made  in  JV,  By  the  method  explained  in  the  last  arti- 
cle, we  can  determine,  for  each  value  of  iV",  the  corresponding 
value   of  X,  either  exactly  or  approximatively. 

The  value  of  x,  corresponding  to  any  assumed  value  of  the 
number  JV,  is  called  the  logarithm  of  that  number ;  and  a  is 
called  the  base  of  the  system  in  which  the  logarithm  is  taken. 
Hence, 

The  logarithm  of  a  number  is  the  exponent  of  the  power  to  which 
it  is  necessary  to  raise  the  base,  in  order  to  produce  the  given  number. 
The  logarithms  of  all  numbers  corresponding  to  a  given  base  constitute 
a  system  of  logarithms. 

Any  positive  number  except  1  may  be  taken  as  the  base 
of  a  system  of  logarithms,  and  if  for  that  particular  base,  we 
suppose  the  logarithms  of  all  numbers  to  be  computed,  they 
will  constitute  what  is  called  a  sy stein  of  logarithms.  Hence, 
we  see  that  there  is  an  infinite  n^:mber  }f  systems  of  ioga 
rithms. 


CHAP.    IX.]  THEORY   OF   LOGARIl  flMS.  •      287 

228.  The  base  of  the  common  system  of  logarithms  is  10, 
and  if  we  designate  the  logarithm  of  any  number  taken  in 
that   system   by   log,    we  shall   have, 


(10)0  =     1 

;   whence. 

loa    1=0 

(10)1  =    10 

,   whence, 

log   10  =  1 

(10)2  ^   100 

whence, 

log  100  =  2 

(10)3=  1000^ 

whence. 

log  1000  =  3 

&c.. 

(fee. 

We  see,  that  in  the  common  system,  the  logarithm  of  any 
number  between  1  and  10,  is  found  between  0  and  1.  The 
logarithm  of  any  number  between  10  and  100,  is  between  1  and 
2 ;  the  logarithm  of  any  number  between  100  and  1000,  is  be- 
tween  2   and   3 ;    and   so  on. 

The  logarithm  of  any  number,  which  is  not  a  perfect  power 
of  the  base,  will  be  equal  to  a  whole  number,  ^lus  a  fraction, 
the  value  of  which  is  generally  expressed  decimally.  The  entire 
part  is  called   the   characteristic^  and  sometimes  the  index. 

By  examining  the  several  powers  of  10,  we  see,  that  if  a 
number  is  expressed  by  a  single  figure,  the  characteristic  of  its 
logarithra  will  be  0 ;  if  it  is  expressed  by  two  figures,  the 
characteristic  of  its  logarithm  will  be  1  ;  if  it  is  expressed  by 
three  figures,  the  characteristic  will  be  2  ;  and  if  it  is  expressed 
by  11  places  of  figures,  the   characteristic  will  be  w  —  1. 

If  the  number  is  less  than  1,  its  logarithm  will  be  negative, 
atid  by  considering  the  powers  of  10,  which  are  denoted  by 
negative   exponents,  we   shall   have, 

=  .1 ;  whence,     log    .1        =  —-  1. 

=  .01 ;         whence,     log    .01      =  —  2. 


(10)-' 

= 

1 

10 

(10)-' 

= 

1 

100 

(10)-' 

= 

1 

1000 

&C 

,  &c. 

=  .001 ;       whence,     log    .001    =  —  3. 

&c.,     &c. 

Here  w^e  see  that  the  logarithm  of  every  number  between  1  and 
.1  will  be  found  between  0  and  —  1  ;  that  is,  it  will  be  equal  to 
—  1,  j^liiB  a  fraction  less  than  1.      The  logarithm  of  any  number 


288 


ELEMENTS   OF   ALGEBRA. 


[CHAP.   IX. 


between  .1  and  .01  will  be  between  —1  and  —2;  that  is,  it 
will  be  equal  to  —  2,  plus  a  fraction.  The  logarithm  of  any 
number  between  .01  and  .001,  will  be  between  —  2  and  —  3, 
or  will  be  equal  to  ~  3,  plus  a  fraction,  and  so  on. 

In  the  first  case,  the  characteristic  is  —  1,  in  the  second  —  2, 
fn  the  third  —  3,  and  in  general,  the  characteristic  of  the  logarithm 
of  a  decimal  fraction  is  negative,  and  numerically  1  greater  than 
the  number  of  Qfs  which  immediately  follow  the  decimal  point.  The 
decimal  part  is  always  positive,  and  to  indicate  that  the  negative 
sign  extends  only  to  the  characteristic,  it  is  generally  written 
over  it;  thus, 
log  0.012  =  2.079181,     which  is  equivalent  to     —  2  +  .079181. 

228"^%  A  table  of  logarithms,  is  a  table  containing  a  set  of 
numbers,  and  their  logarithms  so  arranged  that  we  may,  by  its 
aid,  find  the  logarithm  of  any  number  from  1  to  a  given  num- 
ber, generally  10,000. 

The  following  table  shows  the  logarithms  of  the  numbers,  from 
I  to  100. 


N. 

I.O?. 

N. 

nog. 

N. 

Log. 

N. 

Ijop. 

1 

0.000000 

26 

1.414973 

51 

1.707570 

76 

1.880814 

2 

0.301030 

27 

1.431364 

52 

1.716003 

77 

1.886491 

3 

0477121 

28 

1.447158 

53 

1.724276 

78 

1.892095 

4 

0.602060 

29 

1.462398 

54 

1.732394 

79 

1.897627 

5 

0.698970 

30 

1.477121 

55 

1.740363 

80 

1.903090 

6 

0.778151 

31 

1.491362 

56 

1.748188 

81 

1.908485 

7 

0.845098 

32 

1.505150 

57 

1.755875 

82 

1913814 

8 

0,903090 

33 

1.518514 

58 

1.763428 

83 

1.919078 

9 

0.954243 

84 

1.531479 

59 

1.770852 

84 

1.924279 

10 

1.000000 

35 

1.544068 

60 

1.778151 

85 

1.929419 

11 

1.041393 

86 

1.556303 

61 

1.785330 

86 

1.984498 

12 

1.079181 

37 

1.568202 

62 

1.792392 

87 

1.939519 

13 

1.113943 

88 

1.579784 

63 

1.799341 

88 

1.944483 

14 

1.146128 

89 

1.591065 

64 

1.806180 

89 

1,949390 

15 

1.176091 

40 

1.602060 

65 

1.812913 

90 

1.954243 

16 

1  204120 

41 

1.612784 

66 

1.819544 

91 

1.959041 

17 

1.230449 

42 

1.623249 

67 

1.826075 

92 

1.963788 

18 

1.255273 

43 

1.633468 

68 

1.832509 

93 

1.968483 

19 

1.278754 

44 

1.643453 

69 

1.838849 

94 

1.973128 

20 

1.301030 

45 

1.653213 

70 

1.845098 

95 

1.977^24 

21 

1322219 

46 

1.662758 

71 

1851258 

96 

1.982271 

22 

1.342423 

47 

1.672098 

72 

1.857333 

97 

1986772 

23 

1.361728 

48 

1.681241 

73 

1.863323 

98 

1991226 

24 

1380211 

49 

1.690196 

74 

1.869232 

99 

1.995635 

25 

1.397940 

50 

1.698970 

75 

1.875061 

100 

2.000000 

CHAP.  IX.]       THEORY  OF  LOGARITHMS.  289 

When  the  number  exceeds  100,  the  characteristic  of  its  loga- 
rithm is  not  written  in  the  table,  but  is  always  known,  since 
it  is  1  less  than  the  number  of  places  of  figures  of  the  given 
number.  Thus,  in  searching  for  the  logarithm  of  2970,  in  a  table 
of  logarithms,  we  should  find  opposite  2970,  the  decimal  part 
.472756.  But  since  the  number  is  expressed  by  four  figures, 
the  characteristic  of  the  logarithm  is  3.     Hence, 

log  2970  =  3.472756, 
and  by  the  definition   of  a  logarithm,  the  equation 
a"  =z  iV,     gives 

103.472756  _  2970, 

General  Properties  of  Logarithms, 

229.  The  general  properties  of  logarithms  are  entirely  inde- 
pendent of  the  value  of  the  base  of  the  system  in  which  they 
are  taken.  In  order  to  deduce  these  properties,  let  us  resume 
the   equation, 

in    which    we  may  suppose    a   to   have  any  positive  value    ex- 
cept 1. 

230.  If,  now,  we  denote  any  two  numbers  by  N'  and  Jf'^ 
and  their  logarithms,  taken  in  the  system  whose  base  is  a, 
by  x'  and  a/^,  we  shall  have,  from  the  definition  of  a  logarithm, 

a^'  =W (1), 

and,  a^"=W (2). 

If  we  multiply  equations  (1)  and  (2)  tcgether,  member  liy 
member,  we  get, 

^x'+x-  -  iV"/  X  W'    -    .     -     (3). 

But  since  a  is   the  base   of   the   system,    we   have  from  the 

definition, 

^  ^ocf'  ^  log  {N'  X  W') ;     that  is, 

The  logarithm   of  the  ^product  of  two   numbers  is  equal  to   the 

sum  of  their  logarithms, 

19 


290  ELEMENTS   OF  ALGEBRA?  [CHAP.    IX 

231.  If  we   divide   equation  (1)  by  equation   (2),  member   by 
member,  we   liave, 

""     =-w^ w 

But,  from  the  definition, 


aK-a/^  =  log^-^j;     that  is, 


The  logarithm  of  the   quotient  which  arises  from  dividing  one*' 
number  hy  another  is  equal  to  the  logarithm  of  the  dividend  minus 
the  logarithm  of  the  divisor, 

232.  If  we  raise  both  members  of  equation  (1)  to  the  n^^ 
power,  we   have, 

f^nx'  ^   jsf^^ (5). 

But  from  the  definition,  we  have, 

nx'  =2  log  (iV''")  ;     that  is. 

The  logarithm  of  any  power  of  a  number  is  equal  to  the 
logarithm  of  the  number  multiplied  hy  the  exponent  of  the  power, 

233.  If  we  extract  the  ti*^  root  of  both  members  of  equation 
(1),  we  shall  have, 

z'  1 

a"   z:z{Ny=:   \fW     -      .       (6). 

But  from  the  definition, 

—  =  log  (\/F) ;     that  is, 

The  logarithm  of  any  root  of  a  number  is  equal  to  the  loga* 
rithm  of  the  number   divided  by  the  index   of  the  root. 

234#  From  the  principles  demonstrated  in  the  four  preceding 
articles,  we   deduce  the  following  practical  rules : — 

First,  To  multiply  quantities  by  means  of  their  logarithms. 

Find  from  a  table,  the  logarithms  of  the  given  factors,  take 
the  sum  of  these  logarithms,  and  look  in  the  table  for  the  i?or- 
responding   number;    Ms   will  be    the  product  required. 


CHAP.  IX.J       THEORY  OF  LOGARITHMS.  291 

Thus,  log    7 0.845098 

log    8 0.903090 

log  50 1.748188 ; 

hence,  7  x  8  =  56. 

Second.  To   divide  quantities  by  means  of  their  logarithms. 

Find  the  logarithm  of  the  dividend  and  the  logarith7n  of  the 
divisor,  from  a  table  ;  subtract  the  latter  from  the  former,  and 
look  for  the  number  corresponding  to  this  difference ;  this  will  be 
the   quotient   required. 

Thus,  log  84    - 1.924279 

log  21 1.322219 

log    4 0.602060 ;  ' 

hence,  27  "^  ^* 

Third,  To  raise   a  number  to   any  power. 

Find  from  a  table  the  logarithm  of  the  number,  and  multiply  it 
by  the  exponent  of  the  required  power ;  find  the  number  corres- 
ponding to  this  product,  and  it  will  be  the  required  power. 

Thus,  log    4 0.602060 

3^ 

log  64 1.806180 ; 

hence,  (4)^  =  64. 

Fourth,  To   extract  any  root  of  a  number. 

Find  from  a  table  the  logarithm  of  the  number,  and  divide 
this  by  the  index  of  the  root ;  find  the  number  correspcmding  to 
this  quotient,  and   it  will   be   the  root   required. 

Thus,  log  64 1.806180(6 

log     2 .301030; 

f. 

hence,  ^/^64  =  2. 

By  the  aid  of  these  principles,  we  may  write  d^vQ  following 
equivalent   expressions : — 


292  ELEMENTS    OF  ALGEBRA,     *  [CHAP.    IX. 

Log     {a  .b  .  t  ,  d ,  ,  .  ,)  =.  log  a  -\-  log  b  +  log  c  .  .  . . 

Log     (-T- 1  =  log  a  +  log  6  +  log  c  —  log  0?  —  log  «. 

Log  (a'" .  i^ .  cP  . . . .  )  =  m  log  a  +  n  log  6  +  ^  log  c  +  . . . . 

Log  {a?  —  x^)  =z  log  (a  +  a:)  +  log  (a  —  x). 

Log  y  (a2  -  x^)  =ilog[a-\-  x)  +^ log  (a  —  x). 

Log  (a3  X  1/^)  =  3f  log  a. 

234.  We  have  already  explained  the  method  of  determining 
the  characteristic  of  the  logarithm  of  a  decimal  fraction,  in  the' 
common  system,  and  by  the  aid  of  the  principle  demonstrated 
in  Art.  231,  we  can  show 

That  the  decimal  part  of  the  logarithm  is  the  same  as  the  decimal 
part  of  the  logarithm  of  the  numerator,  regarded  as  a  whole  number. 

For,  let  a  denote  the  numerator  of  the  decimal  fraction,  and 
let  m  denote  the  number  of  decimal  places  in  the  fraction,  then 
will   the   fraction   be   equal    to 

a 

and  its   logarithm   may   be   expressed   as   follows : 

^^g   10^  =  ^^g  ^  "■  ^^§  (1^)""  =  H  «  -  mlog  10  =  log  a-m, 

but  m  is  a  whole  number,  hence  the  decimal  part  of  the  loga 
rithm  of  the  given  fraction  is  equal  to  the  decimal  part  of 
log  a,  or  of  the  logarithm  of  the  numerator  of  the  given 
fraction. 

Hence,  to  find  the  logarithm  of  a  decimal  fraction  from  the 
common   table, 

Lo?Jc  for  the  logarithm  of  the  number,  neglecting  the  decimal 
point,  and  then  prefix  to  the  decimal  part  found  a  negative  charao- 
teristic  equal  to  1  more  than  the  number  of  zeros  which  immediately 
follow  the  decimal  point  in  the  given  decimal. 

The  rules  given  for  finding  the  characteristic  of  the  logarithms 
taken  in  the  common  system,  will  not  apply  in  any  other 
system,    nor   could   we   find   the   logarithm  of  decimal  fractions 


CHAP.  IX.:       THEORY  OF  LOGARITHMS.  2&3 

directly  from  the  tables  in  any  other  system  than  that  whose  base 
is    10. 

These  are  some  of  the  advantages  which  the  common  system 
possesses  over  every  other  system. 

235.  Let   us   again   resume   the   equation 

a»  =  jsr. 

1st.  If  we  make  ]V=1,  x  must  be  equal  to  0,  since  a^  ^  I  ; 
that  is, 

The  logarithm  of  1  in  any  system  is  0. 

2d.  If  we  make  J!i  =:  a,  x  must  be  equal  to  1,  since  a^  =^a- 
that  is, 

Whatever  be  the  base  of  a  system^  its  logarithm^  taken  in  that 
system^  is  equal  to   1. 

Let   us,  in   the   equation, 

a*  =  iv; 

First^   suppose     a  >  1. 

Then,  when  N=  1,  a;  =  0;  when  iV'>  1,  a;  >  0  ;  when  iV<  1, 
jc  <  0,  or  negative ;   that  is. 

In  any  system  whose  base  is  greater  than  1,  the  logarithms  of 
all  numbers  greater  than  1  are  positive^  those  of  all  numbers  less 
than  1  are  negative. 

If  we   consider  the  case  in  which   i\r<  1,   we   shall   have 

a-^  =  iV,      or      —  =  K, 

Now,  if  N  diminishes,  the  corresponding  values  of  x  must 
increase,  and  when  N  becomes  less  than  any  assignable  quan- 
tity, or  0,  the   value  of  x  must  be  C30 :    that  is, 

The  logarithm  of  0,  in  a  system  whose  base  is  greater  than  I, 
is  equal  to  -—  od. 

Second,   suppose     a  <  L 

Then,  when  iV=  1,  x=zO',  when  iV<  1,  ^^  >  0 ;  wheniV'>l, 
r  <  0,  or  negative  :    that  is, 


294   ^  ELEMENTS   OF   ALGEBRA.  [CHAP.    IX 

In  any  system  whose  hose  is  less  than  1,  the  logarithms  of  all 
numbers  greater  than  1  are  negative,  and  those  of  all  numbers  less 
tha7i   1  are  positive. 

If  we  consider  the 'case  in  which  iV<  1,  we  shall  have  a*  =  iV, 
in  which,  if  iV  be  diminished,  the  value  of  x  must  be  increased ; 
and  finally,  when  JV  =zO,  we  shall  have  x  =z  co:   that  is, 

The   logarithm  of  0,  in  a   system   whose   base  is  less  than  1,  is 

ef/2ial  to  -\-  00. 

Finallj,  whatever  values  we  give  to  x,  the  value  of  a*  or 
N  will  always  be  positive;  whence  we  conclude  that  negative 
numbers   have   no   logarithms, 

Logarithm.ic  Series. 

236.  The   method  of  resolving   the   equation, 

a'  =z  6, 

explained  in  Art.  226,  gives  an  idea  of  the  construction  of  loga- 
rithmic tables ;  but  this  method  is  laborious  when  it  is  necessary 
to  approximate  very  near  the  value  of  x.  Analysts  have  dis- 
covered much  more  expeditious  methods  for  constructing  new 
tables,  or  for  verifying  those  already  calculated.  These  methods 
consist  in  the  development  of  logarithms  into  series. 
If  w^e   take   the   equation, 

a^  =  y, 

and  regard   a   as  the  base  of  a  system  of  logarithms,  we  shall 

have, 

log  y  =:X. 

The  logarithm  of  y  will  depend  upon  the  value  of  y,  and 
also  upon  a,  the  base  of  the  system  in  which  the  logarithms 
are   taken. 

Let  it  be  required  to  develop  log  y  into  a  seiies  arranged 
according  to  the  ascending  powers  of  y,  with  co-efficients  that 
are  independent  of  y  and  dependent  upon  a,  the  base  of  the 
system. 


CHAP.   IXJ  LOGARITHMIC   SERIES.  295 

Let   US   first  assume   a  development  of  the  required  form, 

log  y  =  if  +  iVy  +  Py2  +  ^2^3  _,.  &c., 

in  which  if,  iV,  P,  &;c.  are  independent  of  y,  and  dependent 
upon  a.  It  is  now  required  to  find  such  values  for  these  co- 
efficients as  will  make  the  development  true  for  every  value 
of  y. 

Now,  if  we  make  y  =  0,  log  y  becomes  infinite,  and  is  either 
negative  or  positive,  according  as  the  base  a  is  greater  or  less 
than  1,  (Arts.  234  and  235).  But  the  second  member  under 
this  supposition,  reduces  to  if,  a  finite  number  :  hence,  the 
development  cannot  be   made   under   that  form. 

Again,  assume, 

log  y  =  My  +  Ny'^  +  Py^  +  &c. 

If  we  make  y  =  0,  we   have 

log  0  =  0     that  is,     ±00  =  0, 

which  is  absurd,  and  therefore  the  development  cannot  be  made 
under   the   last  form.     Hence,  we  conclude  that, 

T 

The   logarithm   of  a   number    cannot    he   developed  according   to  . 
the   ascending  powers  of  that  number. 

Let    us    write    (1  +  y),   for   y   in    the   first    member   of    the 
assumed   development;    we   shall   have, 

log  (1  +  y)  =  ify  +  W  +  Py^  +  Qy'  +  &c.    .    -    (1), 

making  y  =  0,  the  equation  is  reduced  to  log  1=0,  which  does 
not  present   any   absurdity. 

Since  equation  (1)  is  true  for  any  value  of  y,  we  may  write 
z  for  y;   whence, 

log  {l-\-z)-Mz+  Nz^  +  Pz'^  +  Qz^  +  &e.   -    -    -     (2). 

Subtracting  equation  (2)  from  equation  (1),  member  from  mem- 
oer,  we  obtain, 

iog  (I  +  y)  -  log  (1  +  ^)  =  M{y  ^z)  +  W{y^  -  z'')  -f  P(y3  -  z^) 

+  Q{y'  -  ^0       -   -       -    (3). 


296  ELEMENTS   OF  ALGEBRit.  [CHAP.    IX. 

The  second  member  of  this  equation  is  divisible  by  {y  —  z)y 
let  us  endeavor  to  place  the  .first  member  under  such  a  form 
that  it  shall  also  be  divisible  by  [y  —  z).     We  have, 

log  (1  +  y)  -  log  (1  +  ^)  -  log  (}^)  =  log  (i  +  |-3i| 

But  since  can  be  regarded  as  a  single  quantity,  we  may 

substitute  it  for  y  in  equation  (1),  which  gives, 

Substituting  this  development  for  its  equal,  in  the  first  member 
of  equation  (3),  and  dividing  both  members  of  the  resulting 
equation  by  (y  —  ^),  and   we   have, 

+  Piy''  +  yz  +  ^2)  +  &c. 

Since  this  equation  is  true  for  all  values  of  y  and  0,  m^ke 
z  =^y,  and  there  will  result 

M 
=  if  +  2Ny  +  ZPy'^  +  4§y3  +  5%*  +  &c. 

•*■  ~"  y 

Clearing  of  fractions,  and  transposing,  we   obtain, 


^M+    M 


y-V^P 

+  2iV^ 


2/2 +  4^ 
+  3P 


+  46 


y4+  &c.  =0, 


and   since   this   equation   is   identical,  we   have, 

M  —    J/  =  0 ;     whence,     M  =  M; 

M 
2]Sr+    if  =  0 ;     whence,     iV  =  -  li ; 

3P+2iV^=0;     whence,     P=:-i^=~; 

o  o 

4§  +  3P=ll;     whence,     Q  =  -  ^  =  _  ^. 
&<»  &c. 


CHAP.   IXJ  LOGARITHMIC  SERIES.  297 

The   law   of  the   co-efficients   m   the    development  is   evident; 

M 

the   co-efficient  of  y"    is    qp  — ,  according   as  n  is   even   or  odd. 

Substituting   these   values   for  iV,  P,   Q^  &c.,  in  equation  (1), 
we   find   for   the   development  of  log   (1  +  y)  ; 

/       log  (1  +  y)  =  -%  -  -^  y^  +  g-y^  —  -J  y*  .  .  &o. 

3  ,,4  fl/5 


=  4_fH-C-5+»^.  .*..).  .(4. 


which  is   called  the   logarithmic   series. 

Hence,  we  see  that  the  logarithm  of  a  number  may  be 
developed  into  a  series,  according  to  the  ascending  powers  of 
a  number  less   than   it   by    1. 

In  the  above  development,  the  co-efficients  have  all  been  de- 
termined in  terms  of  M,  This  should  be  so,  since  M  depends 
upon  the  base  of  the  system,  and  to  the  base  any  value  may  be 
assigned.     By  examining   equation  (4),  we  see  that. 

The  expression  for  the  logarithm  of  any  number  is  composed  of 
two  factorSy  one  dependent  on  the  number,  and  the  other  on  the 
base  of  the  system  in  which  the  logarithm  is  taken. 

The  factor  which  depends  on  the  base,  is  called  the  modulus 
of  the   system  of  logarithms. 

237t  If  we  take  the  logarithm  of  \  +  y  in  a  new  system 
and   denote   it   by   ^  (1  -f  y),  we  shall   have, 

Z(l+y)=.lf'(y-|+^-^  +  ^-&c.).   -    (5), 

in  which  M'   is  the  modulus  of  the  new  system. 

If  we  suppose  y  to  have  the  same  value  in  equations  (4)  and  (5), 
and  divide  the  former  by  the  latter,  member  by  member,  we  have 
log  (1  +  y)  _M^ 

/(l+y)-if' 

Z  (1  +  y) :  log  (1  +  y)  : :  i/' :  Jf ;    hence, 
The  logarithms  of  the  same  number,  taken  in  two  different  systemsj 
are  to  each  other  as  the  moduli  of  those  systems. 


-  ,    whence,  (Art.  183,) 


298  ELEMENTS   OF  ALGEBRA.  LCHAP.    IX. 

238 i  Having  shown  that  the  modulus  and  base  of  a  system 
of  logarithms  are  mutually  dependent  on  each  other,  it  follows, 
that  if  a  value  be  assigned  to  one  of  them,  the  corresponding 
ralue  of  the  other  must  be  determined   from  it. 

If  then,  we  make  the  modulus 

M  =1, 
fhe  base  of  the  system  will   assume  a  fixed  value.     The  system 
of  logarithms  resulting  from  such  a  modulus,  and  such  a  base,  is 
called  the  Naperian  System,      This   was   the  first  system  known, 
and  was  invented  by  Baron  Napier,  a  Scotch  mathematician. 

If  we  designate  the  Naperian  logarithm  by  Z,  and  the  loga- 
rithm in  any  other  system  by  log,  the  above  proportion  becomes, 

l{\+y)  :  log(l+y)  :    :  1  :  if ; 
whence,  M  xl{\  +  ?/)  =  log  (1  +  y). 

Hence,  we  see  that. 

The  JSfaperian  logarithm  of  any  number y  multiplied  by  the  modu- 
lus of  any  other  system,  will  give  the  logarithm  of  the  same  number 
in  that  system. 

The  modulus  of  the  Naperian  System  being  1,  it  is  found  most 
convenient  to  compare  all  other  systems  with  the  Naperian ;  and 
hence,   the  modulus  of  any  system   of  logarithms,  is 

The  number  by  which  if  the  Naperian  logarithm  of  any 
mimber  be  multiplied,  the  product  will  be  the  logarithm  of  the 
same  number  in  that  system, 

239.  Again,  M  x  1(1  +  y)  =  \og{\  +  y\    gives 

/(l+,)  =  l2i(^);     that  is, 

The  logarithm  of  any  number  divided  by  the  modulus  of  its 
system,  is  equal  to  the  Naperian  logarithm  of  the  same  number, 

240.  If  we  take  the  Naperian  logarithm  and  make  y  =  I 
equation  (5)   becomes. 


CHAP,    IX.]  LOGARITHMIC   SERIES.  299 

a  series  which  does  not  converge  rapidly,  and  in  which  it  would 
be  necessary  to  take  a  great  number  of  terms  to  obtain  a  near 
approximation.  In  general,  this  series  will  not  serve  for  deter- 
mining the  logarithms  of  entire  numbers,  since  for  every  number 
greater  than  2  we  should  obtain  a  series  in  which  the  terms 
would   go    on   increasing   continually. 

241 1  In  order  to  deduce  a  logarithmic  series  sufficiently  con 
verging  to  be  of  use  in  computing  the  Naperian  logarithms 
of  numbers,  let  us  take  the  logarithmic  series  and  make 
M'^  1.  Designating,  as  before,  the  Naperian  logarithm  by  /j  we 
shall   have, 

/(l+y)=2,-|  +  |-^+|!-&c.   .--    (1). 

If  now,  we   write   in   equation   (1),   —  y  for  y,  it  becomes, 

,a-,)=-,-|-|-|-|-*e...,.) 

Subtracting  equation  (2)  from  (1),  member  from  member, 
we   have, 

l{\+y)-l{\-y)=^2(y  +  t+t^   l!+^  +  &c.)--  (3). 

But, 
/(I  +  y)  -l{\-y)  =  l  (^^)  ;      whence, 

If  now   we  make = ,  we  shall  have, 

I —y  2    '  ' 

(1  +  y)^  =  (1  -7j)  {z  +  1),     whence,     y  =  gj^ry.' 

Substituting  these  values  in  equation  (4),  and  observing  that 
{^-~)  =  K^  +  1)  -13    we  find, 


800  ELEMENTS   OF  ALGI^BRA.  [CHAP.   IX. 

li.  +  1)-  /.  =  2(^  +  3^^3  +  ^^,+  &c.)(5), 
or,  by  trajnsposition, 

Let  us  make  use  of  formula  (6)  to  explain  the  method  of 
computmg  a  table  of  Naperian  logarithms.  It  may  be  remarked, 
that  it  is  only  necessary  to  compute  from  the  formula  the 
logarithms  of  prime  numbers;  those  of  other  numbers  may  be 
found  by  taking   the   sum  of  the   logarithms  of  their  factors. 

The  logarithm  of  1  is  0.  If  now  we  make  ^  =  1,  we  can 
find  the  logarithm  of  2 ;  and  by  means  of  this,  if  we  make 
0  =  2,  we  can  find  the  logarithm  of  3,  and  so  on,  as  exhibited 
below. 

n=0     .     .     - =0.000000; 

^3  =  0.693147  +  2  (i  +  3^- +  ^  +  ^  ...)=  1.098612 

?4  =  2x^:2 =1.386294 

Z5  =  1.386294  +  2(-i  +  3l^+^,  +  ^...)==  1.609437 

/6  =  Z2+Z3 =1.791759 

n  =  1.791759  +  2  (1  +  3_-L_  +  ^, +  ...)  =  1.945910 

/8  =  Z4  +  Z2 =2.079441 

Z9=2x/3 .' =2.197224 

n0=/5  +  ^2 =2.302585 

&;c.  &c. 

In  like  manner,  we  may  compute  the  Naperian  logarithms 
©f  all   numbers.      Other   formulas   may   be   deduced,  which  are 


CHAP.   IX.]  LOGAEITHMIO  SERIES.  801 

more  rapidly  :onverging  than  the  one  above  given,  but  this 
serves  to  sho\i  the  facility  with  which  logarithms  may  be  com- 
puted, 

241*.  We  have  already  observed,  that  the  base  of  the  common 
system  of  logarithms  is  10.  We  will  now  find  its  modulus. 
We  have, 

l{l  +y)  :  log  (l  +  y)  :   ^,  1:M  (Art.  238). 

If  we  make  y  =  9,  we   shall  have, 

nO:       log  10       :    :       1  :  if. 

But  the  no  =  2.302585093,     and    log  10=1  (Art.  228); 

hence,      31  =     qaoxq^aoo  =  0.434294482  =  the  modulus  of  the 

common  system. 

If  now,  we  multiply  the  Naperian  logarithms  before  found,  by 
this  modulus,  we  shall  obtain  a  table  of  common  logarithms 
(Art.  238). 

All  that  now  remains  to  be  done,  is  to  find  the  base  of  the 
Naperian  system.  If  we  designate  that  base  by  e,  we  shall  have 
(Art.  237), 

le  :  loge  :    :  1  :  0.434294482. 

But     le  z=l  (Art.  235)  :    hence, 

1  :  log  e  :    :  1  :  0.434294482 ; 

hence,  log  e  =  0.434294482. 

But  as  we  have  already  explained  the  method  of  calculating 
the  common  tables,  we  may  use  them  to  find  the  number  whose 
logarithm  is  0.431294482,  which  we  shall  find  to  be  2.718281828  ; 
hence, 

c  =  2.718281828 

We  see  frcm  the  last  equation  but  one,  that 

The  modulus  of  the  common  system  is  equal  to  the  common  loga 
rithm  of  the  Naperian  base. 


S02  ELEMENTS   OF  ALGEBRA.  [CHAP.    IX 

Of  Interpolation, 

242.  When   the  law  of  a   series   is   given,  and   several   term* 

taken   at   equal    distances    are    known,    we   may,   by  means   of 

the   formula, 

^                  _    .    nin  —  l')  .    .   n(n  —  1)  (ti  —  2)  ,    .    „  ,,^ 

T=a-\~nd,  +  -A_-^cf,  -{-  -A ^V _;^^  +  &c.  -  -  -.  (I), 

already  deduced,  (Art.  209),  introduce  other  terms  between 
them,  which  terms  shall  conform  to  the  law  of  the  series 
This   operation    is   called   interpolation. 

In  most  cases,  the  law  of  the  series  is  not  given,  but  only 
numerical  values  of  certain  terms  of  the  series,  4aken  at  fixed 
intervals ;  in  this  case  we  can  only  approximate  to  the  law 
of  the  series,  or  to  the  value  of  any  intermediate  term,  by 
the   aid   of  formula  (1). 

To  illustrate  the  use  of  formula  (1)  in  interpolating  a  terni 
in  a  tabulated  series  of  numbers,  let  us  suppose  that  we  have 
the  logarithms  of  12,  13,  14,  15,  and  that  it  is  required  to  find 
the  logarithm  of  \2\-.  Forming  the  orders  of  differences  from 
the  logarithms  of  12,  13,  14  and  15  respectively,  and  taking 
the  first  terms  of  each, 

12  13  14  15 

1.079181,  1.113943,  1.146128,  1.176091, 

0.034762,  0.032185,  0.029963, 

-  0.002577,  -  0.002222, 

+  0.000355, 
we  fmd      d,  =.  0.034762,  d,z=,  -  0.002577,  d,  :=.  0.000355. 

If  we  consider  log  12  as  the  first  term,  we  have  also 

a  =  1.079181     and     n  = -i-- 

Making  these  several  substitutions  in  the  formula,  and  no- 
glecting  the  terms  after  the  fourth,  since  they  are  inappieciable 
we  find, 


CHAP.    IX.J  FORMULAS    FOR  INTEREST.  803 

or,   by    substituting   for   d^,  d^,  &;c.,  their  values,  and   for   a   its 
value, 

a 1.079181 

^d, 0.017381 

^cfa     -        -                 ....  0.000322 

^\d^ 0.000022 

Log  12^                 -        -        .        .  1.096906 

Had  it  been  required  to  find  the  logarithm  of  12.39,  we 
should  have  made  n  =  .39,  and  the  process  would  have  been 
the  same  as  above.  In  like  manner  we  may  interpolate  terms 
between  the   tabulated  terms  of  any  mathematical  table. 


INTEREST. 

243 •  The  solution  of  all  problems  relating  to  interest,  may 
be  greatly  simplified  by  employing  algebraic  formulas. 

In  treating  of  this  subject,  we  shall  employ  the  following 
notation : 

Let  p  denote  the  amount  bearing  interest,  called  the  principal  ; 
r        "       the  part  of  $1,  which  expresses   its  interest  for 

one  year,  called  the  rate  per  cent; 
t        "       the  time,  in   years,  that  p   draws  interest; 
t         "       the  interest  of  p  dollars  for  t  years ; 
S        "      J5  +   the  interest  which   accrues   in  the  time    L 

This  sum  is  called  the  amount. 

Simple  Interest  . 

To  find  the  interest  of  a  sum  p  for  t  years^  at  the  rate  r,  and 
the   amount   then   due. 

Since  r  denotes  the  part  of  a  dollar  which  expresses  its  in- 
terest for  a   single  year,  the  interest  of  p   dollars  for  the  same 


S04  ELEMENTS   OF   ALGEBRA.  [CHAP.    IX. 

time  wUl  be  expressed  "by  jpr ;  and  for  t  years  it  will  be  t  timei 
as   much :    hence, 

i^Vi^ (1); 

and  for  the  amount  due, 

5  =  2? +i?ifr  =^(1  +  ^r)     -    .     (2). 

EXAMPLES. 

1.  What  is  the  interest,  and  what  the  amount  of  $365  for  three 
fears  and  a  half,  at  the  rate  of  4  per  cent,  per  annum.     Here, 

^  =  $365 ; 

^  =  4  =  0.04; 

«  =  3.5 ; 

i  =ptr  =  365  X  3.5  X  0.04  =  $51,10 : 
hence,  ^  =  365  +  51,10  =  $416,10. 

Present   Value  and  Discount  at  Simple  Interest 

The  present  value  of  any  sum  S^  due  t  years  hence,  is  the  prm- 
cipal  j9,  which  put  at  interest  for  the  time  t^  will  produce  the 
amount  ;S^. 

The  discount  on  any  sum  due  t  years  hence,  is  the  difference 
between  that  sum  and  the  present  value. 

To  find  the  'present  value  of  a  sum  of  dollars  denoted  by  S,  due 
i  years  hefice,  at  simple  interest,  at  the  rate  t;  also,  the  discount. 

We  have,  from  formula  (2), 

S  =p  +  ptr-y 

and  since  p  is  the  principal  which  in   t  years  will  produce  the 
sum  aS^,  we  have. 


CHAP.   IX.]  FORMULAS  FOR  INTEREST.  805 

and  for  the  discount,  which  we  will  denote  by  D,  we  have 

n  =  s-^-^-=^^^  .  .  (4). 

^      ^       l  +  tr       V^tr  ^  ^ 

1.  Required  the  discount  on  $100,  due  3  months  hence,  at  the 
rate  of  5^  per  cent,  per  annum. 

S  =  $100         =  $100, 

t  =  3  months  =  0.25. 

.=^  =.055.- 

Hence,  the  present  value  p  is 

hence,  i)  =>S  -  ^  =  100  -  98,648  =  $1,357. 

Compound  Interest. 

Compound  interest  is  when  the  interest  on  a  sum  of  money 
becoming  due,  and  not  paid,  is  added  to  the  principal,  and 
the  interest  then  calculated  on  this  amount  as  on  a  new 
principal. 

To  find  the  amount  of  a  sum  p  placed  at  interest  for  t  years, 
compound  interest  being  allowed  annually  at  the  rate  r. 

At  the  end  of  one  year  the  amount  will  be, 

S  =  p  +  pr  =  p(l  +  r). 

Since  compound  interest  is  allowed,  this  sum  now  becomea 
the  principal,  and  hence,  at  the  end  of  the  second  year,  the 
amount   will   be, 

S'  =zp{\  +  r)  '\-pr{\  +  r)  =  p{l  +  r)^. 

Eegard  ^(1  +  t*)^  as  a  new  principal ;  we  have,  at  the  end 
of  the   third   year, 

S''  =p{l+rY+pr{l  +  rY=zp{l  +  r)3; 

20  ^J^. 


306  ELEMENTS  OF  ALGEBRA.  [CHAP.   IX, 

aid   at   the  end  of  t  years, 

^  =  ^(l  +  r)^     ....    (5). 
And  from  Articles  230    and  232,  we  have, 

log  S  =±  logp  +  t  log  (1  -f-  r)  ; 

and  if  any  three  of  the  four  quantities  S,  p,  t,  and  r,  are  given^ 
the  remaining  one  can  be  determined.  ^ 

Let  it  be  required  to  find  the  time  in  which  a  sum  p  will 
double  itself  at  compound  interest,  the  rate  being  4  per  cent, 
per  annum. 

We  have,  from  equation  (5), 

S  =p{l  +  ry. 

But  by  the  conditions  of  the  question, 

S=2p=p{l  +  ry: 

hence,  2  =  (l+r)^ 

__       log  2       _  0.301030 
^^  ^  ""  log  (1  +  r)  ""  0.017033' 

=  17.673  years, 

=  17  years,  8  months,  2  days, 

To  find  the  Discount. 

The  discount  being  the  difference  between  the  sum  S  and  p^ 
we  have. 


V 


CHAPTER  X. 


GENERAL  THEORY  OF  EQUATIONS. 


244.  Every  equation  containing  but  one  unknown  quantity 
which  is  of  the  m*^  degree,  m  being  any  positive  whole  number, 
may,  by  transposing  all  its  terms  to  the  first  member  and  divid- 
ing by  the  co-efficient  of  x^,  be  reduced  to  the  form 

xm  +  p^m-1  ^    Qxv^2  ^     ^     ^     ^     ^    ^   2'x  +    U  =  0, 

In  this  equation  P,  Q,  .  ,  .  ,  T,  U,  are  co-efficients  in  the 
most  general  sense  of  the  term ;  that  is,  they  may  be  positive 
or  negative,  entire  or  fractional,  real  or  imaginary. 

The  last  term  U  is  the  co-efficient  of  a;®,  and  is  called  the 
absolute  term.   ^ 

If  none  of  these  co-efficients  are  0,  the  equation  is  s;aid  to  be 
corriplete ;  if  any  of  them  are  0,  the  equation  is  said  to  be 
incomplete,  , 

In  discussing  the  properties  of  equations  of  the  m*^  degree, 
involving  but  one  unknown  quantity,  we  shall  hereafter  suppose 
them  to  have  been  reduced  to  the  form  just  given. 

245.  We  have  already  defined  the  root  of  an  equation  (Art.  77) 
to  be  ani/  expression^  which^  when  substituted  for  the  unknown 
quantity  in   the  equation,  will  satisfy  it. 

We  have  shown  that  every  equation  of  the  first  degree  has 
one  root,  that  every  equation  of  the  second  degree  has  two 
roots;  and  in  general,  if  the  two  members  of  an  equation  are 
equal,  they   must  be    so    for    at    least    some    one   value   of   the 


308  *  ELEMENTS   OF   ALGEBRA.  [CHAP.   X. 

unknown  quantity,  either  real  or  imaginary.  Sucti  value  of  the 
unknown  quantity  is  a  root  of  the  equation :  hence,  we  infer,  that 
every  equation,  of  whatever  degree,  has  at  least  one  root. 

We  shall  now  demonstrate  some  of  the  principal  properties 
of  equations  of  any  degree  whatever. 

First  Projperty, 
246  •  In  every  equation  of  the  form 

if  a  is  a  root,  the  first  member  is  divisible  by  x  -—  a ;  and  con 
versely,  if  the  first  member  is  divisible  by  x  —  a,  a  is  a  root  of 
iiie  equation. 

Let  us  apply  the  rule  for  the  division  of  the  first  member 
by  X  —  a,  and  continue  the  operation  till  a  remainder  is  found 
which  is  independent  of  x ;   that  is,  which  does  not  contain  x. 

Denote  this  remainder  by  R  and  represent  the  quotient  found 
by   Q\  and  we  shall  have, 

Now,  since  by  hypothesis,  a  is  a  root  of  the  equation,  if  we 
substitute  a  for  x,  the  first  member  of  the  equation  will  reduce  to 
zero  ;  the  term  Ql(x  —  a^  will  also  reduce  to  0,  and  consequently, 
we  shall  have 

i^  =  0. 

But  since  R  does  not  contain  x,  its  value  will  not  be  affected 
by  attributing  to  x  the  particular  value  a :  hence,  the  remainder 
R  is  equal  to  0,  whatever  may  be  the  value  of  x,  and  conse- 
quently,  the  first  member  of  the  equation 

^m  +  p^m-l  _}.    g^m-2      ,     ,     ^     ,     J^  Tx -{-   TJ  :=^^, 

is  exactly  divisible  by  x  —  a. 

Conversely,  if  a;  —  a  is  an  exact  divisor  of  the  first  member 
of  the  equation,  the  quotient  Q'  will  be  exact,  and  w^e  shall  have 
/2  =z  0 :   hence, 

^m  ^  pa;»»-i   .   ,       ^    ^Tx-^  U=:  Q'{x  -  a). 


CHAP.  X.]  THEORY  OF  EQUATIONS.  809 

If  now,  we  suppose  a;  ==  a,  the  second  member  will  reduce  to 
zero,  consequently,  the  first  will  reduce  to  zero,  and  hence  a  will 
be  a  root  of  the  equation  (Art.  245).  It  is  evident,  from  the 
nature  of  division,  that  the  quotient   Q'  will  be  of  the  form 

^m-l  ^  p'xm-2 -\-  R'X+    U'  ^0. 

247.  It  follows  from  what  has  preceded,  that  in  order  to  di.-* 
cover  whether  any  polynomial  is  exactly  divisible  by  the  bino- 
mial a?  -—  a,  it  is  sufficient  to  see  if  the  substitution  of  a  for  ^ 
will   reduce   the   polynomial   to   zero. 

Conversely,  if  any  polynomial  is  exactly  divisible  by  x  —  «, 
then  we  know,  that  if  the  polynomial  be  placed  equal  to  zero, 
a  will  be   a   root    of  the   resulting   equation. 

The  property  which  we  have  demonstrated  above,  enables  us 
to  diminish  the  degree  of  an  equation  by  1  when  we  ki;ow 
one  of  its  roots,  by  a  simple  division ;  and  if  two  or  m^re 
roots  are  known,  the  degree  of  the  equation  may  be  still  further 
diminished   by  successive  divisions. 

EXAMPLES. 

1.  A  root  of  the   equation, 

x^  —  25^2  ^  50^  _  36  :^  0, 
is  3 :    what  does  the  equation  become  when  freed  of  this    C6t  ? 
x^  —  25:r2  +  60^  —  36  lb—  3 
x^—    3i;3  aj3  4-3:i'2  — 16.r-|    12. 

-f    3a;3  — 25a;2 
3a:3  _    9^2 


16^2  _^  60:?; 
16.^2  +  48a- 


\2x  -  36 
I2x  -  36 

Ans.  x"^  +  3;z^2  _  io.r  -r  12  ~  0. 
2.  Two   roots   of  the   equation, 

x^  —  12a;3  +  48a;2  —  68a;  +15  =  0, 
are   3   and   5 :    what   does   the    equation   become  when  fre»jd  ^4 
them  1  Ans,  x'^  ^  4iX  -\-  \  z=z  Q 


SIO  ELEMENTS   OF   ALGEBRA.  [CHAP.    X. 

3.  A    root   of  the   equation, 

a;3  -6x^-{-  11a; -6  =  0, 
is   1  :    what   is   the   reduced   equation? 

Ans.   a;2  —  5a?  +  6  =  0. 

4  Two   roots   of  the   equation, 

4:x^  —  Ux^  —  5a?2  -\- Six  -\- 6  =z  0, 
are   2   and   3 :    find   the   reduced   equation. 

Ans.  4x^  +  Qx+  I  :=  0. 

Second  Projjert?/, 

248»  Every  equation  involving  hut  one  unknown  quantity,  has 
xs  many  roots  as  there  are  units  in  the  exponent  which  denotes 
its   degree,    and  no   more. 

Let  the   proposed   equation  be 

^m  ^  p^m-l  _|_   Q^m-2  +    ,    ,    ,    -\.  Tx  +    U  =  0. 

Since  every  equation  is  known  to  have  at  least  one  root 
{Art.  245),  if  we  denote  that  root  by  a,  the  first  member  will 
be   divisible   by  x  —  a,  and  we  shall  have  the  equation, 

But  if  we   place, 

we  obtain   a   new  equation,  which   has   at   least   one   root. 

Denote   this  root  by  b,  and  we    have  (Art.  246), 

^mr-i  _f.  p/^m-2  _j.  _  .  3=  (a;  —  5)  (a;^-2  +  F^'x"^^  +...)• 

Substituting  the  second  member,  for  its  value,  in  equation 
(1),  we   have, 

;j.m  _|.  p^n^-l  4.  _  .  _-  (^  _  «)  (.^.  _  ^,)  (a;m~2  4.  p// ^«-3  -f-  .  .  .)  .  .  (2). 

Reasoning   upon   the   polynomial, 

^m-2  _|_  p//^m-3  +   .    .    ., 

as    upon    the   preceding   polynomial,  we   have 

^m-^  +  P^'x"*-^  +  .  .  .  =z{x  —  c)  (a;^-3  4.  p///^m-^  +...)» 
and   by   substitution, 
igiN,^p^m-l.j.  _     3,,  (^  _  ^)  (^  _  ^j  (^  _  c)  {x'^^  +  F'^'x''^)  - . .  (3), 


CHAP.  X.I         THEORY  OF  EQUATIONS.  311 

By  continuing  this  operation,  we  see  that  for  each  binomial 
factor  of  the  first  degree  with  reference  to  x,  that  we  separate, 
the  degree  of  the  polynomial  factor  is  reduced  by  1  ;  therefore, 
after  m  —  2  binomial  factors  have  been  separated,  the  polynomial 
factor  will  become  of  the  second  degree  with  reference  to  a?, 
which  can  be  decomposed  into  two  factors  of  the  first  degree 
(Art.  115),  of  the  form  x  —  k,   x  —  L 

Now,  supposing  the  m  —  2  factors  of  the  first  degree  to  have 
already  been  indicated,  we  shall  have   the  identical  equation, 

a;m  _^  p^^m^i  4-  .  .  z={x  —  a){x  —  b){x  —  c),.{x  —  k){x'-l)=zO; 

from  which  we  see,  that  the  Jirst  member  of  the  proposed  equation 
may  he  decomposed  into  m  binomial  factors  of  the  first  degree. 

As  there  is  a  root  corresponding  to  each  binomial  factor  of 
the  first  degree  (Art.  246),  it  follows  that  the  m  binomial  factors 

of  the  first  degree,  x  —  a^  x  —  by  x  —  c ,  give  the  m  roots, 

a,  &,  c  .  .  .,  of  the  proposed  equation. 

But  the  equation  can  have  no  other  roots  than  a,  6,  c  .  .  .  ^,  /. 
for,  if  it  had  a  root  a\  different  from  a,  &,  c  .  .  .  .  Z,  it  would 
have  a  divisor  x  —  a\  different  from  x  —  a^  x  —  b,  x  —  c...x—l, 
which   is   impossible ;    therefore. 

Every  equation  of  the  m*^  degree  has  m  roots,  and  can  have 
no   more, 

249.  In  equations  which  arise  from  the  multiplication  of  equal 
factors,  such  as 

{x  -  ay  {x  -  by  {x  -  cY  (x-^d)  =  0, 

the  number  of  roots  is  apparently  less  than  the  number  of  units 
in  the  exponent  which  denotes  the  degree  of  the  equation.  But 
this  is  not  really  so  ;  for  the  above  equation  actually  has  ten 
roots,  four  of  which  are  equal  to  cr,  three  to  b,  two  to  c,  and 
one   to   d. 

It  is  evident  that  no  quantity  a'',  different  from  a,  b,  c,  c?, 
can  verify  the  equation ;  for,  if  it  had  a  root  a^,  the  first  menCfc- 
ber  would  be  divisible  by  a;  —  a^,  which  is  impossible. 


S12  ELEMENTS   OF   ALGEBRA.  [CHAP.    X. 

Consequence  of  the  Second  Property, 

250.  It  has  been  shown  that  the  first  member  of  every  equa- 
tion of  the  w^*'^  degree,  has  m  binomial  divisors  of  the  first 
degree,  of  the   form 

a;  —  a,     x  —  h^     x  —  c^  ,  ,  ,  x  —  k^     x  —  L 

If  we  multiply  these  divisors  together,  two  and  two,  three  and 
three,  &;c.,  we  shall  obtain  as  many  divisors  of  the  second, 
third,  &c.  degree,  with  reference  to  x,  as  we  can  form  different 
combinations  of  m  quantities,  taken  two  and  two,  three  and  three, 
&c.      Now,    the  number  of  these  combinations  is   expressed   by 

m.— ^— ,     m.— ^— .-^—  .  .  .  (Art.  132); 

hence,  the   proposed   equation    has 

m  —  1 


2 

divisors 

of  the 

second   degree 

? 

m  — 

1      m  — 

2 

m.-    g 

'       3 

divisors 

of  the 

third   degree  ; 

m  —  1     1 

m  —  2 
3       * 

m  — 
4 

3 

m.       ^^       . 

divisors  of  the  fourth  des^ree  : 

and   so 

on. 

Composition  of  Equations, 

251,  If  we  resume  the  identical  equation  of  Art.  248, 
^m_|_p,^m-.i  _j_  Qx"^"^  .,,  4-  U  ={x—a)(x  ^h){x  —  c)  .  .  .{x—  I),,. 
and  suppose  the  multiplications  indicated   in  the  second  member 
to  be  performed,  we  shall   have,  from   the   law   demonstrated  ia 
article  135,  the  following  relations : 
F=z--a  —  b  —  c  —  ,,,—k—l,  or  —  P  =  a+b  +  c-\-    ..  -f^^-f^, 

Q  =  ab  -\-  ac  -\- be  + ak  -^  kl, 

B  =z  ^  abc  —  abd  —bed  ...  —  iki,  cr  —  B  =abc  +  abd  -{-...+  iki, 

U=  dz  abed  ....  ikl^  or  ±  11=  abc  ,  ,  ,  ikl 


JHAP.   X.]  COMPOSITION    OF   EQUATIONS.  .S13 

The  double  sign  has  been  placed  before  the  product  of  a,  6,  c,  &c. 
in  the  last  equation,  since  the  product  — ax  —  b  X  —  c  .  .  x  —l^ 
will  be  plus  when  the  degree  of  the  equation  is  even,  and  minus 
when  it  is  odd. 

By  considering  these  relations,  we  derive  the  following  conclu- 
sions with  reference  to  the  values  of  the  co-efficients : 

1st.  The  co-efficient  of  the  second  term,  with  its  sign  changed,  is 
equal  to  the  algebraic  sum  of  the  roots  of  the  equation, 

2d.  The  co-efficient  of  the  third  term  is  equal  to  the  sum  of  the 
different  products  of  the  roots,  taken  two  in  a  set. 

3d.  The  co-efficient  of  the  fourth  term,  with  its  sign  changed,  is 
equal  to  the  sum  of  the  different  products  of  the  roots,  taken  three 
in  a  set,  and  so  on, 

4th.  The  absolute  term,  with  its  sign  changed  when  the  equation 
IS  of  an  odd  degree,  is  equal  to  the  continued  product  of  all  the 
roots  of  the  equation. 

Consequences, 

1.  If  one  of  the  roots  of  an  equation  is  0,  there  will  be 
no  absolute  term ;  and  conversely,  if  there  is  no  absolute  term, 
Dne  of  the  roots  must  be  0. 

2.  If  the  co-efficient  of  the  second  term  is  0,  the  numerical 
sum  of  the  positive  roots  is  equal  to  that  of  the  negative  roots. 

3.  Every  root  will  exactly  divide  the  absolute  term. 

It  will  be  observed  that  the  properties  of  equations  of  the 
ivccond  degree,  already  demonstrated,  conform  in  all  respects  to 
the  principles  demonstrated  in  this  article. 

EXAMPLES    OF    THE    COMPOSITION    OF    EQUATIONS. 

1.  Find  the  equation  whose  roots  are  2,  3,  5,  and  —  6. 

We  have,  from  the  principles  already  established,  the  equation 

whence,  by  the  application  of  the  preceding  principles,  we  obtain 
the  equation, 

a;4  _  4^3  _  29:^2  +  I56x  -  180  =  0. 


314  ELEMENTS   OF   ALGEBRA.  [CHAP.   X 

2.  What  is  the  equation  whose  roots  are  1,  2,  and  —3? 

Ans.   a:^  —  7a;  +  6  =  0. 


Ji.  What  is  the   equation   whose  roots  are  3,    —  4,   2  +  V3, 
nd    2  -^3"?  Ans.   a;^  -  3a:3  -  I6x^  +  49a;  -  12  =  0. 

4.  What  is  the  equation  whose  roots  are  3+y^,  3  — -/s", 
and  —  6?  ^W5.   a;^  —  32a;  +  24  =  0. 

5.  What  is  the  equation  whose  roots  are  1,  -- 2,  3,  —4,  5, 
and  —  6  ? 

Ans.    x^  +  3a;5  -  41a;*  -  87a;3  +  400a;2  +  444a;  -  720  =  0. 

6.  What  is  the  equation  whose  roots  are  ....  2  +  V  ■—  1 , 
2  -y^^^,   and    -  3  ]  Ans.    x'^  -x^ -Ix  +  \b  ^^ 

Greatest  Common  Divisor. 

252.  The  principle  of  the  greatest  common  divisor  is  of  fre- 
quent application  in  discussing  the  nature  and  properties  of 
equations,  and  before  proceeding  further,  it  is  necessary  to  inves- 
tigate a  rule  for  determining  the  greatest  common  divisor  of  two 
or  more  polynomials. 

The  greatest  common  divisor  of  two  or  more  polynomials  is 
the  greatest  algebraic  expression,  with  respect  both  to  co-efficients 
and   exponents,  that  will  exactly  divide   them. 

A  polynomial  is  prime,  when  no  other  expression  except  1 
will   exactly  divide   it. 

Two  polynomials  are  prime  with  respect  to  each  other,  when 
they  have   no   common  factor   except  1. 

253.  Let  A  and  B  designate  any  two  polynomials  arranged 
with  reference  to  the  same  leading  letter,  and  suppose  the 
polynomial  A  to  contain  the  highest  exponent  of  the  leading 
letter.  Denote  the  greatest  common  divisor  of  A  and  B  by  i>, 
and  let   the   quotients  found  by  dividhfg  each  polynomial  by  D 


CHAP.   X.]  GREATEST  COMMON"  DIVISOR.  815 

be  represented  by  A^  and  B^  respectively.     We  shall  then  have 
the  equations, 

^=.A',     and    ^=B; 

whence,  A  =:  A^  X  I>    and     B  =:  B^  X  D. 

Now,  B  contains  all  the  factors  common  to  A  and  B,  For, 
if  it  does  not,  let  us  suppose  that  A  and  B  have  a  common 
factor  d  which  does  not  enter  i>,  and  let  us  designate  the  quo- 
tients of  A^  and  B\  by  this  factor,  by  A^^  and  B^\  We  shall 
then  have, 

A  =  A'\d.J)    and    B  =  B'\d.D', 

or,  by  division, 

"^    =:  A''    and    -A^  =  B'\ 


d.B'^  d.D 

Since  A^^  and  B"  are  entire,  both  A  and  B  are  divisible  by 
d  .  i),  which  must  be  greater  than  i>,  either  with  respect  to  its 
co-efficients  or  its  exponents  ;  but  this  is  absurd,  since,  by 
hypothesis,  D  is  the  greatest  common  divisor  of  A  and  B. 
Therefore,    D  contains  all   the   factors   common   to  A   and  B, 

Nor  can  D  contain  any  factor  which  is  not  common  to  A 
and  B,  For,  suppose  D  to  have  a  factor  d^  which  is  not  con 
tained  in  A  and  jB,  and  designate  the  other  factor  of  D  by  i>' ; 
we   shall   have  the   equations, 

A  =  A\d\D'     and    B^B^d'.D"', 

or,    dividing   both   members   of  these   equations   by  d\ 

^=zA\D'    and    ^  =  B\D'. 
w  d^ 

Now,  the  second  members  of  these  two  equations  being  en- 
tire, the  first  members  must  also  be  entire ;  that  is,  both  A 
and  B  are  divisible  by  d^  and  therefore  the  supposition  that 
d^  is   not   a   common  factor  A  A  and  B  is  absurd.     Hence, 

1st.  The  greatest  common  divisor  of  two  polynomials  contains 
all  the  factors  common  to  the  polynoitials,  and  does  not  contain 
any  other  factors. 


816  ELEMENTS   OF  ALGEBKA.  [CBAF.   X, 

254.  If,  now,  we  applj  the  'rule  for  dividing  A  by  i?,  and 
continue  the  process  till  the  greatest  exponent  of  the  leading 
letter  in  the  remainder  is  at  least  one  less  than  it  is  in  the 
polynomial  B,  and  if  we  designate  the  remainder  by  E,  and 
the    quotient  found,  by   Q,  we    shall   have, 

A  =  Bx  Q  +  B    -    -    -    -    (1). 

If,  as  before,  we  designate  the  greatest  common  divisor  of 
A  and  B  by  D,  and  divide  both  members  of  the  last  equatiou 
by  it,  we   shall   have. 

Now,  the  first  member  of  this  equation  is  an  entire  quantity, 

and    so   is   the   first   term   of    the   second   member ;    hence     ~ 

must  be  entire;   which  proves  that  the  greatest  common  divisor 
of  A   and  B  also   divides  E. 

If  we  designate  the   greatest   common  divisor  of  B  and  E  by 
i)^,  and  divide  both  members  of  equation  (1)  by  it,  we  shall  have, 
A  _B      -,E 

Now,  since  by  hypothesis  D^  is  a  common  divisor  of  B  and 
i?,  both  terms  of  the  second  member  of  this  equation  are 
entire ;  hence,  the  first  member  must  be  entire ;  which  proves 
that  the  greatest  common  divisor  of  B  and  E^  also  divides  A. 

We  see  that  D\  the  greatest  common  divisor  of  B  and  E^ 
cannot  be  less  than  i>,  since  D  divides  both  B  and  i^ ;  nor  can 
i>,  the  greatest  common  divisor  of  A  and  B^  be  less  than  D\ 
because  D^  divides  both  A  and  B ;  and  since  neither  can  be  less 
than  the  other,  they  must  be  equal ;  that  is,  B  =  D',     Hence, 

2d.  The  greatest  common  divisor  of  two  polynomials,  is  the  same 
as  that  betiveen   the   second  polynorr^ial  and    their  remainder   after 


From  the  principle  demonstrated  in  Art.  253,  we  see  that  wo 
may  multiply  or  divide  one  polynomial   by   any    factor    that   is 


OHAP.   X.1  GREATEST  COMMON  DIVISOR.  317 

not  contained  iu  the  other,  without  jiffecting  their  greatest   com- 
mon divisor. 

255,  From  the  principles  of  the  two  preceding  articles,  we 
deduce,  for  finding  the  greatest  common  divisor  of  two  poly- 
nomials, the  following 

RULE. 

j.  Suppress  the  monomial  factors  common  to  all  the  terms  of  the 
first  polynomial ;  do  the  same  with  the  second  polynomial ;  and  if 
the  factors  so  suppressed  have  a  common  divisor^  set  it  aside,  as 
forming  a  factor  of  the  common  divisor  sought. 

II.  Prepare  the  first  polynomial  in  such  a  manner  that  its  first 
term  shall  be  divisible  by  the  first  term  of  the  second  polynomial, 
both  being  arranged  with  reference  to  the  same  letter :  Apply  the 
rule  for  division,  and  continue  the  process  till  the  greatest  exponent 
of  the  leading  letter  in  the  remainder  is  at  least  one  less  than  it  is 
in  the  second  polynomial.  Suppress,  in  this  remainder,  all  the 
factor's  that  are  common  to  the  co-efficients  of  the  different  powers 
of  the  leading  letter ;  then  take  this  result  as  a  divisor  and  the 
second  poly  normal  as  a  dividend,  and  proceed  as  before. 

III.  Continue  the  operation  until  a  remainder  is  obtained  which 
will  exactly  divide  the  preceding  divisor  ;  this  last  remainder,  mul- 
iiplied  by  the  factor  set  aside^  will  be  the  greatest  common  divisor 
sought;  if  no  remainder  is  found  which  will  exactly  divide  the 
preceding  divisor,  then  the  factor  set  aside  is  the  greatest  coramon 
divisor  sought. 

EXAMPLES. 

1.  Find  the  greatest  common  divisor  of  the  polynomials 
a3  __  a?j)  +  3a62  _  3^3^      ^nd      a^  —  bab  +  4&2. 

First   Operation.  Second   Operation, 


a}  —  a?b  +  8a62  _  353 


Aa%  -    ab^  -  363 


22       5a6  +  4Z>2 


a  +46 


1st  rem.  VJab^  —  I9b^ 
or,  1962  (a -6). 

Hence,  c  —  6  is  the  greatest  common  divisor. 


a2  _  ^cib  -f  462 


—  4a6  -f-  462 


\a  -  b 


a  .-4& 


0. 


318  ELEMENTS   OF  ALGEBRA.  [CHAP.   X. 

We  begin  by  dividing  the  polynomial  of  the  highest  degree 
by  that  of  the  lowest ;  the  quotient  is,  as  we  see  in  the  above 
table,  a  +  4i,  and  the  remainder  lOai^  —  1953^ 

But,  19aZ»2  -  19^,3  ^  1952  (a  _  5) 

Now,  the  factor  lOi^^  will  divide  this  remainder  without  dividing 
a?  -  bab  +  452 : 
hence^  th3  factor  must  be  suppressed,  and  the  question  is  reduced 
to  finding  the  greatest  common  divisor  between 
a^  —  bah  ■\-  4^2      and      a  —  b. 

Dividing  the  first  of  these  two  polynomials  by  the  second,  there 
is  an  exact  quotient,  a  —  46 ,  hence,  a  —  b  is  the  greatest  com- 
mon divisor  of  the  two  given  polynomials.  To  verify  this,  lei; 
each  be   divided   by  a  —  b. 

3.  Find   the   greatest   common   divisor  of  the   polynomials, 
8a5  —  5a^^  +  2a¥     and       2a^  —  Za?b'^  +  b\ 

We  first  suppress  a,  which  is  a  factor  of  each  term  of  the 
first   polynomial :    we  then   have, 

3a^  -  baW  +  2¥  ||  2a*  -  ^a^"^  +  b\ 

We  now  find  that  the  first  term  of  the  dividend  will  not  con- 
tain the  first  term  of  the  divisor.  We  therefore  •  multiply  the 
dividend  by  2,  w^hich  merely  introduces  into  the  dividend  a 
factor  not  common  to  the  divisor,  and  hence  does  not  affect 
the   common  divisor   sousjht.     We   then   have, 


!2a^  -  3a262  _|.  54 


—         a2^2_|_      ^4 

_  o2  {a?  -  62). 
We  find    after    division,  the    remainder  —  o?b'^  -\-  6*    which  wo 
put   under   the   form     —  ^2  (^^2  _  i;zy      ^y^   ^j^^.^   suppress    —  b\ 
and   divide. 


2a*  -  3a262  +  6* 

1  a2  -  62 

2a*  -  2^262 

2a2  -^  62 

—  aW  +  6* 

—  a262  -f  b\ 

Hence,    a^  —  6^    is   the   greatest   cc  mmon    divisor. 


CHAP.   X.J  GKEATEST   COMMON   DIVISOR.  319 

3.  Let  it  be   required   to   find    the    greatest   common   divisor 
between   the  two   polynomials, 

—  3^3  +  Sab^  —  aH  +  a^,      and      46^  -  5ab  +  a\ 
First  Operation, 


—  12Z>=^ 

+  12a52  -  4a^  + 

4a3 

4^2  _  5^J  _(.  «2 

!«t  rem. 

-    - 

-  Sab^~    a%-\-    4a3 

—  12a62  -  ^a^b  -{-  16a3 

-  36,     -  3a 

2d  rem. 
or, 

—  \^a?b  4 
19a2(- 

Second   Ope 
462  _  5^5  +  «2 

-  19a^ 

a). 

—    ab4-a^ 

-•46  +  a 

0. 

Hence,    —  6  +  a,    or    a  —  b^    is  the  greatest  common  divisor 

In  the  first  operation  we  meet  with  a  difficulty  in  dividing  the 

two  polynomials,  because   the  first  term  of  the   dividend   is   not 

exactly  divisible  by   the   first   term   of  the   divisor.     But   if  wo 

observe  that  the  co-efficient  4,  is   not   a  factor  of  all  the  terms 

of  the   polynomial 

462  _  5^5  _|.  ^2^ 

and   therefore,  by  the  first  principle,  that  4   cannot  form  a  part 

of  the   greatest   common   divisor,  we    can,  without  afiecting   this 

common   divisor,  introduce   this  factor   into   the   dividend.     This 

gives, 

-  1263  +  12a62  -  4a26  +  4a3, 

and   then   the   division   of  the   terms   is   possible. 

Effecting    this    division,    the   quotient   is     —  36,    and    the   re 

mainder  is, 

—  3a62  —  a?b  +  4a3. 

As   the    exponent   of    6    in   this    remainder    is   still   equal  to 

that   of   6    in   the   divisor,    the   division    may   be   continued,   b;)? 

multiplying  this  remainder  by  4,  in  order  to  render  the  division 

of  the  first   term   possible.     This  done,  the   remainder  becomes 

-  12a62-4a26  -MGa^; 


820  ELEMENTS   OF   ALGEBRA.  [CHAP.   X 

which,  divided  by  4b^  —  5ab  +  a'^,  gives  the  quotient  —  3a, 
which  should  be  separated  from  the  first  by  a  comma,  having 
no    connexion  with   it.     The   remainder   after   this   division,  is 

-  19a26  +  19a3. 

Placing  this  last  remainder  mider  the  form  lOa^  (_  5 -}- a), 
and  suppressing  the  factor  19a^,  as  forming  no  part  of  the  com- 
mon divisor,  the  question  is  reduced  to  finding  the  greatest 
common   divisor   between 

AP  —  bah  +  a^     and     —h  +  a. 

Dividing  the  first  of  these  polynomials  by  the  second,  we 
obtain  an  exact  quotient,  —  46  +  a :  hence,  —  5  +  a,  or  a  —  b, 
is   the   greatest   common   divisor    sought. 

256.  In  the  above  example,  as  in  all  those  in  which  the 
exponent  of  the  leading  letter  is  greater  by  1  in  the  dividend 
than  in  the  divisor,  we  can  abridge  the  operation  by  first  mul- 
tiplying every  term  of  the  dividend  by  the  square  of  the  co- 
effieieixt  of  the  first  term  of  the  divisor.  We  can  easily  see 
that  by  this  means,  the  first  term  of  the  quotient  obtained  will 
contain  the  first  power  of  this  co-efficient.  Multiplying  the 
divisor  by  the  quotient,  and  making  the  reductions  with  the 
dividend  thus  prepared,  the  result  will  still  contain  the  co-eflicient 
as  a  factor,  and  the  division  can  be  continued  until  a  remainder 
is  obtained  of  a  lower  degree  than  the  divisor,  with  reference 
to   the   leading   letter. 

Take   the   same   example  as  before,  viz. : 

—  363  _^  3^52  __  a^  +  a3     and     462  _  5^5  ^  a\ 
and  multiply  the  dividend  by  4^  =  16 ;    and  we   have 


First    Operation, 
~  4863  +  48^52  _  lQa^  +  16^3 


-  12a62  —    4a26  +  16a3 


1462  -  5ab  -I-  a 


U  n^ 


-  126  -  3a 


1st  remainder,  —  19a26  -f-  19a^ 

or,  19a2  (-  6  +  «)• 


CHAP.  X.]  GREATEST  COMMON  DIVISOR.  821 


Second    Opemtion. 


Al)^  —  bah  -f-  o? 


ab  +  o? 


-    b  +  a 


—  46  + a 


2i1  remainder,  —  0. 

When  the  exponent  of  the  leading  letter  in  the  dividend 
exceeds  that  of  the  same  letter  in  the  divisor  by  two,  three, 
&;c.,  multiply  the  dividend  by  the  third,  fourth,  &;c.  power  of 
the  co-efficient  of  the  first  term  of  the  divisor.  It  is  easy  to 
see   the   reason  of  this.  ' 

257.  It  may  be  asked  if  the  suppression  of  the  factors,  com 
mon  to  all  the  terms  of  one  of  the  remainders,  is  absolutely 
necessary^  or  whether  the  ol)ject  is  merely  to  render  the  opera- 
tions more  simple.  It  will  easily  be  perceived  that  the  suppres- 
sion of  these  factors  is  necessary ;  for,  if  the  factor  lOa^  was  not 
suppressed  in  the  preceding  example,  it  would  be  necessary  to 
multiply  the  whole  dividend  by  this  factor,  in  order  to  render 
its  first  term  divisible  by  the  first  term  of  the  divisor ;  but, 
then,  a  factor  would  be  introduced  into  the  dividend  which  is 
also  contained  in  the  divisor ;  and,  consequently,  the  required 
greatest  common  divisor  would  contain  the  factor  lOa^  whicli 
should  form  no  part  of  it. 

258.  For  another  example,  let  it  be  required  to  find  the 
greatest  common  divisor  of  the  two  polynomials, 

a*  +  Za%  +  Aa?b'^  —  6«53  +  2b^     and     Aa^  +  2ab^  —  263, 
or  simply  of, 

a*  +  Sa^  +  4a262  -  Qab^  +  2b^     and     2a^  +  a6  —  b^, 
since   the   factor   2b   can    be   suppressed,    being   a   factor   of  the 
second  polynomial  and  not  of  the  first. 


First    Operation. 
Sa*  +  2ia^  +  S2a^^  -  ASab^  +  16b^ 


-f-  20a^  -f  36a262  _  48a63  +  166* 


2a2  +  a5  —  62 
4a2  +  10a6  +  ^36* 


+  26^262  -  38a63  -f  166* 

_ A _ 

i St  remainder,  — 51a63 -f  296* 

or,  -  63(51a  -  296). 


822  ELEMENTS   OF   ALGEBRA.  'CHAP.   X, 

Second    Operation, 

Multiply  by  2601,  the  square  of  51. 

5202a2  +  2m\ah  -  2601^2     ||    5ia  -    29i 


5202a2  —  2958a5  102a  +  1096 

1st  remainder,         +  5559a6  -  2601^2 
5559a5- 316162 


2d  remainder,  -f    56062. 

The  exponent  of  the  letter  a  in  the  dividend,  exceeding  that 
of  the  same  letter  in  the  divisor,  by  two^  the  whole  dividend 
is  multiplied  by  2^  =  8.  This  done,  we  perform  the  division^ 
and  obtain  for  the   first  remainder, 

-  51a63  +  296^ 

Suppressing  —  6^,  this  remainder  becomes  51a  —  296 ;  and 
the  new   dividend  is 

2a2  +  a6  —  62. 

Multiplying  the  dividend  by  (51)2  _  2601,  then  effecting  the 
division,  we  obtain  for  the  second  remainder  +  56062.  Now,  it 
results  from  the  second  principle  (Art.  254),  that  the  greatest 
common  divisor  must  be  a  factor  of  the  remainder  after  each 
division;  therefore  it  should  divide  the  remainder  56062.  g^j|; 
this  remainder  is  independent  of  the  leading  letter  a  :  hence,  if 
the  two  polynomials  have  a  common  divisor,  it  must  be  inde- 
pendent of  a,  and  will  consequently  be  found  as  a  factor  in  the 
co-efficients  of  the  different  powers  of  this  letter,  in  each  of  the 
proposed  polynomials.  But  it  is  evident  that  the  co-efficients  of 
these  powers  have  not  a  common  factor.  Hence,  the  tivo  given 
polynomials  are  prime  with  respect  to  each  other, 

259.  The  rule  for  finding  the  greatest  common  divisor  of  two 
polynomials,  may  readily  be   extended    to   three   or   more   poly 
nomials.     For,  having   the   polynomials  A,  B^   (7,  7>>,  &c.,  if  we 
fmd    the  greatest   common   divisor   of  A   and   B^   and   then   the 
iireatest  common  divisor  of  this  result  and  (7,  the  divisor  so  ob 


CHAP.   X.]  GREATEST   COMMON  DIVISOR.  328 

tained  will  evidently  be  the  greatest  common  divisor  of  A,  B^ 
and  (7;  and  the  same  process  may  be  applied  to  the  remaining 
polynomials. 

260.  It  often  happens,  after  suppressing  the  monomial  factoit* 
common  to  all  the  terms  of  the  given  polynomials,  and  arrangin<» 
the  remaining  polynomials  with  reference  to  a  particular  letter, 
that  there  are  polynomial  factors  common  to  the  co-efficients  of 
the  different  powers  of  the  leading  letter  in  one  or  both  poly- 
nomials. In  that  case  we  suppress  those  factors  in  both,  and  if 
the  suppressed  factors  have  a  common  divisor,  we  set  it  aside,  as 
forming  a  factor  of  the  common  divisor  sought. 

} 

EXAMPLE. 

Let  it  be  required  to  find  the  greatest  common  divisor  of  the 
two  polynomials 

a?d'^  ■—  <^d?  —  oi^c^  +  c*,     and     ^dj^d  —  2ac^  +  2c^  —  4acd, 

The  second  contains  a  monomial  factor  2.  Suppressing  it, 
and  arranging  the  polynomials  with  reference  to  d,  we  have 

(a2  _  c2)  d"-  —  a^c^  +  c\     and     (2a2  —  2ac)  d  —  ac^  +  cK 

By  considering  the  co-efficients,  a^  —  c^  and  —  a^c^  -f-  c*,  in  the 
first  polynomial,  it  will  be  seen  that  —  a^c^  +  c^  can  be  put  under 
the  form  — €"^{0?  —  c^):  hence,  a^  —  c^  is  a  common  factor  of  the 
co-efficients  in  the  first  polynomial.  In  like  manner,  the  co-effi- 
cients in  the  second,  2d^  —  2ac  and  — -  ac^  +  c^,  can  be  reduced 
to  2a(a  —  c)  and  —  c^{a  —  c)  ;  therefore,  a  —  c  is  a  common 
factor  of  these  co-efficients. 

Comparing  the  two  factors  a^  —  c^  and  a  —  c,  we  see  that  the 
las^.  will  divide  the  first;  hence,  it  follows  that  a  — c  is  a  com- 
mon factor  of  the  proposed  polynomials,  and  it  is  therefore  a 
factor  of  the  greatest  common  divisor. 

Suppressing  a^  —  ^2  {^  the  first  polynomial,  and  a  —  c  in  the 
second^  we  obtain   the  two  polynomials, 

c?2  —  c2      and      2ad  —  ^^ 


324  ELEMENTS   OF   ALGEBRA,  ICHAP.    X, 

to    wnich   the   jrdiiiary  process  maj   be   applied. 


4a2f^2  _  4^2c2 


2ad  ■ 


2ad  +  c2 


+  2ac'^d  —  ^a^c^ 


After  having  multiplied  the  dividend  by  Aa^^  and  performed 
the  division,  we  obtain  a  remainder  •—  4a2c2  -f-  c*,  independent  of 
the  letter  d :  hence,  the  two  polynomials,  d"^  —  (?  and  2ad  —  c2, 
are  prime  T^dth  respect  to  each  other.  Therefore,  the  greatest 
common  divisor  of  the  proposed  polynomials  is  a  —  c. 

261.  It  sometimes  happens  that  one  of  the  polynomials  cou 
tains  a  letter  which  is  not  contained  in  the  other. 

In  this  case,  it  is  evident  that  the  greatest  common  divisor  is 
independent  of  this  letter.  Hence,  by  arranging  the  polynomial 
which  contains  it,  with  reference  to  this  letter,  the  required  com- 
mon  divisor  will  be  the  same  as  that  which  exists  between  the  co- 
efficients of  the  different  power's  of  the  principal  letter  and  the 
second  polynomiaL 

Bj  this  method  we  are  led,  it  is  true,  to  determine  the  great 
est  common  divisor  between  three  or  more  polynomials.  But 
they  will  be  more  simple  than  the  proposed  polynomials.  It 
often  happens,  that  some  of  the  co-efficients  of  the  arranged 
polynomial  are  monomials,  or,  that  we  can  discover  by  simple 
inspection  that  they  are  prime  with  respect  to  each  other ;  and. 
\\i  this  case,  we  are  certain  that  the  proposed  polynomials  are 
prime   with   respect   to   each   other. 

Thus,  in  the  example  of  the  last  article,  after  having  suppressed 
tlio  common  factor  a  —  c,  which  gives  the  results, 

c?2  _  ^2      and      2ad  -  c\ 

we  know  immediately  that  these  two  polynomials  are  prime  with 
respect  to  each  other ;  for,  since  the  letter  a  is  contained  in  the 
second  and  not  in  the  first,  it  follows  from  what  has  just  been  said, 
that  the  common  divisor  must  be  contained  in  the  co-efficients  2'i 


CHAP.   X.J  GREATEST  COMMON   DIVISOR.  325 

and  —  c^ ;  but  these  are  prime  with  respect  to  each  other,  and 
consequently,  the  expressions  d?  —  c^and  2ad  —  c^,  are  also  prime 
with  respect  to  each  other. 

Let  it  be  required  to  find  the  greatest  common  divisor  cf  the 
two   polynomials, 

Zhcq  +  oOmp  +  186c   +  bmpq^ 
and,  4,adq  —  42^   +  24a(^  —  "Ifgq. 

Now,  the  letter  b  is  found  in  the  first  polynomial  and  not  in 
the  second.  If  then,  we  arrange  the  first  with  reference  to  h^ 
we  have, 

{Zcq  -4-  18c)  5  +  30mp  +  bmpq, 

and  the  required  greatest  common  divisor  will  be  the  same  as 
that  which  exists  between  the  second  polynomial  and  the  two 
co-efficients  of  b^  w^hich  are, 

^cq  +  18c      and      30mp  +  ^mpq. 

Now,  the  first  of  these  co-efficients  can  be  put  under  the  form 
^c{q  +  6),  and  the  other  becomes  bmp{q  +  6) ;  hence,  g'  +  6  is 
a  common  factor  of  these  co-effipients.  It  will  therefore  be 
sufficient  to  ascertain  whether  q  -\-  Q  is  a  factor  of  the  second 
polynomial. 

Arranging  this  polynomial  with  reference   to  g,  it  becomes 
{4.ad-^lfg)q^A^fg  +  24.ad', 

and  as  the  second  part,  24a cZ  —  42y^  =  Q{Aad  —  "Yfg)^  it  follows 
that  this  polynomial  is  divisible  by  g'  +  6,  and  gives  the  quotient 
4:ad  —  Ifg,  Therefore,  g'  +  6  is  the  greatest  common  divisor  of 
the  proposed  polynomials. 

EXAMPLES. 

1.  Eind  the  greatest  common  divisor  of  the  two  polynomialj 
6x^  —  Ax^  -  lla;3  —  Sx^  -  3a:  —  1, 
and  4z*  -f  2x^  —  18.^2  +   8a;  —  5. 

Ans     2x^  —  4:X^  ^  x  —  \ 


« 


326  ELEMENTS   OF   ALGEBRA.  |CHAF.    X. 

2.  Pii.d   the   greatest   common    divisor  of  the   polynomials 

and  Idx*  —    9a;3 -f  47a;2  —  21a;   -f- 28. 

'    ^?i5.   5a;2  —  3a;  +  4. 

3.  rind  the  greatest   common   divisor  of  the  two  polynomials 

5a*^»2  ^  2a3^3  _|_  ^^2    _  3^2^4  ^  5^^^ 

A71S   a^  -f-  «^. 
Transformation  of  Equations, 

262t  The  object  of  a  transformation,  is  to  change  an  equation 
from  a  given  form  to  another,  from  which  we  can  more  readily 
determine   the   value  of  the  unknown   quantity. 

First, 

To  change  a  given  equation  involving  fractional  co-efficients  to  anothe? 
of  the  same  general  form,  huthaving  the  co-efficients  of  all  its  termsentire 
If  we   have   an  equation  of  the   form 

V 
and    make  x  =.  -r'') 

k 

in-  which  y  is  a  new  unknown  quantity,  and  k  entirely  arbitrary ; 
we  shall  have,  after  substituting  this  value  for  ar,  and  multiplying 
every  teim  by  k^, 

an  equation  in  which  the  co-efficients  of  the  different  powers  of 
y  are  equal  to  those  of  the  same  powers  of  x  in  the  given  equa- 
tion, multiplied  respectively  by  P,  A:i,  P,  k^,  k^,  &c. 

It  is  now  requtred  to  assign  such  a  value  to  k  as  will  make 
the  CO  efficients  of  the  different  powers  of  y  entire. 

To  iixisitrate,  let  us  take,  as  a  general  example,  the  equation 


CHAP    X.]         TRANSFORMATION   OF   EQUATIONS.  327 

wliich  becomes,  after  substituting  -|-  for  x,  and  multiplying  by  k\ 
ak  cTc^  eJc^  glc^ 

Now,  there  may  be  two  cases — 

1st.  Where  the  denominators  6,  d^  /,  A,  are  prime  with  respect 
to  each  other.  In  this  case,  as  k  is  altogether  arbitrary,  take 
k  =  hdfh^  the  product  of  the  denominators^  the  equation  will  then 
become, 

2/4  +  adfh .  y3  +  ch'^dph'^  .  2/2  +  ehH^f-h^  .  y  +  ghH^fh^  =0, 

in  which  the   co-efRcients   of  y  are   entire,  and   that  of  the   first 
term  is  1. 

2d.  When  the  denominators  contain  common  factors,  we  shall 
evidently  render  the  co-efficients  entire,  by  making  k  equal  to  the 
least  common  multiple  of  all  the  denominators.  But  we  can 
simplify  still  more,  by  giving  to  k  such  a  value  that  A?^,  P,  A:^,  .  .  . 
shall  contain  the  prime  factors  which  compose  6,  c?,  /,  A,  raised 
to  powers  at  least  equal  to  those  which  are  found  in  the  de- 
nominators. 

Thus,   the   equation 


becomes 


4  ^      3_i.    ^      2  ^  ^^      -A 

"^    ■~"6"''   ■^12'^   ■"  "150 "^  ""  9000- ^' 


6  ^   "*■   12  ^        150^       9000""    ' 


y 

after  making   x  =  — ,  and  reducing  the  terms. 

First,  if  we  make  k  —  9000,  which  is  a  multiple  of  all  the 
other  denominators,  it  is  clear  that  the  co-efficients  become  entire 
numbers. 

But  if  we  decom^pose  6,  12,  150,  and  9000,  into  their  prime 
factors,  we  find, 

Q  =  2xS,     12  =  22x3,     150  =  2  X  3x52,     9000  =  2"^  K  32  x  5^  • 

and   by  making 

^  =  2  X  3  X  5, 


328  ELEMENTS   OF  ALGEBRA.  [CHAP.  5 

the  product  of  the   different  prime   factors,  we   obtain 

k^  =  2^  XS^X  52,     A-3  ^  23  X  33  X  53,     A;*  =  2*  X  3*  X  5* ; 
■whence    we   see   that    the   values   of    k,   A:^,   P,   k^,   contain  the 
prime   factors   of  2,    3,    5,    raised    to   powers   at   least  equal   to 
those  which   enter  into  6,  12,  150,  and  9000.     Hence,  making 

^'  —  2  X  3  X  5, 
is   sufficient   to   make   the   denominators   disappear.     Substituting 
this  value,  the   equation   becomes 

5.2.3.5    3      5.2^.32.52   ^      7.23.33.53  13.2^.3^.5'^  _ 

^  2.3  .  ^  "^      22.3~~  ^  2.3.52    y         23.32.53     ~     ' 

which  reduces   to 

y^  -  5.5?/3  +  5.3.52^/2  -  7.22. 32. 5y  _  13.2.32.5  =  0  ; 
or,         y^  -  25y3  4-  375y2  __  1260?/  ~  1170  »=  0. 

Hence,  we  perceive  the  necessity  of  taking  k  as  small  a 
number  as  possible  :  otherwise,  we  should  obtain  a  transformed 
equation,  having  its  co-efficients  very  great,  as  may  be  seen  by 
reducing  the  transformed  equation  resulting  from  the  supposi- 
tion  ^  —  9000. 

Having  solved  the  transformed  equation,  and  found  the  values 
of  y,    the   corresponding   values    of  x  may   be   found   from   the 

y 

equation,  x  =:  — -, 

by  substituting  for  y  and  k  their  proper  values. 

EXAMPLES.  I 

1.  x^ x'^  ~\ X =  U- 

3       ^36         72  . 

V 
Making     x  =-^ ,     and  we  have, 

y3_  14?/2  +  lly-75  =  0. 

13   ,   ,  21  32    ,        43  1 

^-         ^^-12^+40^  -225"  -600^-800  ==^- 

Making  X  =  ^-  =  ^,     and  we  have, 

y«  -  65y*  +  1890y3  _  30720?/^  -  928800y  -  972000  =:  0. 


CHAP.  X.] 


TRANSFORMATION  OF  EQUATIONS. 


829 


Second. 

To   make    the    second    or    any   other     term  disappexr    from    an 
equation. 

263«  The  difficulty  of  solving  an  equation  generally  diminishes 
with  the   number   of  terms   involving   the   unknown   quantity. 

Thus   the   equation 

x^  =  3',     gives  immediately,     a;  =  ±  .Vg^ 
while  the   complete   equation 

x^  +  2px  +  q  =  0, 
requires  preparation   before  it  can   be   solved. 

Now,  any  given   equation   can   always   be   transformed  into  an 
incomplete  equation,  in  which  the  second  term  shall  be  wanting. 

For,  let  there  be  the  general  equation. 

Suppose  X  =  u  -\-  x% 

u  being   a  new  unknown  quantity,  and  x'  entirely  arbitrary. 

By  substituting  u  +  ^^  for  x,  we  obtain 
{u  +  x')'^+F{u  +  a/)"^!  +  Q{u  +  x')"^^  ...+T{u  +  x')  +  U=zO. 

Developing  by  the  binomial  formula,  and  arranging  with  refer- 
ence to  u,  we   have 


w^  +  wa/ 

m  —  1    ,^ 
u^-^  +  m .  — - —  ar  ^ 

W^2  ^  ^ 

.  .  +  a/«» 

+  P 

+  -(m-  l)Fx' 

+  p^/^i 

'          +Q     . 

+  . . . 

^  =  0. 

+  Tx' 

+  U 

Since  a/  is   entirely  arbitrary,  we 

may 

dispose  of  it  in   such 

way  that  we  shall  have 

'                   i 

fwo/  +  P  =  0 ;     when 

ce, 

X'^r- 

F 

330  ELEMENTS   OF  ALGEBRA.  LCHAP    X. 

Substituting  this  value  of  x'  in  the  last  equation,  we  shall 
obtain   an   incomplete  equation   of  the   form, 

um  +  Q'u'^^  +  i^/^^s  -\.  ,  ,  ,  T'u+  U'  =  0, 

ill  which   the   second   term   is  wanting. 

If  this  equation  were  solved,  we  could  obtain  any  value  of 
a   corresponding   to   that   of  w,  from   the   equation 

P 

X  =  u  -{-  x\     smce     x  =  u . 

m 

We  have,  then,  in  order  to  make  the  second  term  of  an 
equation   disappear,  the  following 

RULE. 

Substitute  for  the  unknown  quantity/  a  new  unknown  quantity 
minus  the  co-efficient  of  the  second  term  divided  by  the  exponent 
which  expresses    the  degree  of  the  equation. 

Let  us   apply  this   rule   to   the   equation, 

x^  +  2px  =:  q. 
If  we   make  x  =z  u  —  p, 

we  have  {u  —  pY  -\-  2p  {u  —  p)  =  q; 

and   by   performing    the    indicated    operations    and   transposing, 
we  find 

263*.  Instead  of  making  the  second  term  disappear,  it  may 
be  required  to  find  an  equation  which  shall  be  deprived  of  its 
third,  fourth,  or  any  other  term.  This  is  done,  by  making  the 
co-efficient  of  u,  corresponding   to   that   term,    equal   to   0. 

For  example,  to  make  the  third  term  disappear,  we  make, 
in   the   transformed  equation,  (Art.  263), 

m^^—  x'^  +  {m  "l)Fx'  +  Q  =  0, 

from   which   we   obtain   two  values    for   x\  which   substituted   m 
the    transformed  equation,  reduce  it  to   the  form, 

w"»  +  P'w^i  -{-  E'li"^^  ...-{-  T'u+  IP  =0. 


CHAP.   X]  OP  DERIVED   POLYNOMIALS.  331 

Beyond  the  third  term  it  will  be  necessary  to  solve  an 
equation  of  a  degree  superior  to  the  second,  to  obtain  the  value 
of  ic';  and  to  cause  the  last  term  to  disappear,  it  will  be  neces- 
sary to  solve   the   equation, 

which    is   what    the    given    equation   becomes   when   a;'   Is   sub- 
stituted  for   X, 

It  may  happen  that  the  value, 

m 
which  makes  the  second  term  disappear,  causes  also  the  disap 
pearance  of  the  third  or  some  other  term.  For  example,  in 
order  that  the  third  term  may  disappear  at  the  same  time 
with  the  second,  it  is  only  necessary  that  the  value  of  (x/^ 
which  results   from   the   equation, 

^'  =  ~^, 

m 
shall  also  satisfy  the  equation, 
^  _  1 

P 

Now,  if  in  this  last  equation,  we  replace  a/  by ,    we  have 

m  —  1  P2  p2 

and,  consequently,  if 

_  2me 

the  disappearance  of  the   second  term  will  also  involve  that  of 
the   third. 

Formation  of  Derived  JPolynomiah, 

264.  That  transformation  of  an  equation  which  consists  in 
substituting  u  -{'  x^  for  a?,  is  of  frequent  use  in  the  discussion 
of  equations.  In  practice,  there  is  a  very  simple  method  of 
obtaining  the  transformed  equation  which  results  from  this  sub 
sjtitutior^ 


382  ELEMENTS   OF   ALGEBRA.  [CHAP.   X 

To    show  this,  let  us  substitute  for  x,  u  +  x'  in  the  equation 

then,  bj  developing,  and   arranging  the   terms   according   to   th« 
ascending   powers  of  u,  we   have 

m  —  1    ,    „ 


•\-Fx'^ 

-\-qxf^ 

+  .   . 
'\rTx' 


+  .  .  . 


w+wi- 


1.2 


W2  + 


^=0. 


Bj  examining  and  comparing  the  co-efficients  of  the  diiierent 
powers  of  u^  we  see  that  the  co-efficient  of  w^,  is  what  the  first 
member  of  the  given  equation  becomes  when  x'  is  substituted 
in  place  of  x-^  we  shall  denote  this   expression  by  X\ 

The  co-efficient  of  u^  is  formed  from  the  preceding  term  X'^ 
by  multiplying  each  term  of  X  by  the  exponent  of  xf  in  that 
term,  and  then  diminishing  this  exponent  by  1  ;  we  shall  denote 
this  co-efficient  by  Y\ 

The  co-efficient  of  v?  is  formed  from  Y\  by  multiplying  each 
term  of  Y'  by  the  exponent  of  xf  in  that  term,  dividing  the 
product  by  2,  and  then  diminishing  each  exponent  by  1.     Repre- 

senting  this  co-efficient  by  -— ,  we  see  that  Z'  u  formed  from  J^, 

in  the   same  manner  that  Y^  is  formed  from  X\ 

In  general,  the  co-efficient  of  any  power  of  t/,  in  the  above 
transformed  equation,  may  be  found  from  the  preceding  co-efficient 
in  the  following  manner,  viz. : — 

Multiply  each  term  of  the  preceding  co-efficient  by  the  exponent 
of  xf  in  that  term^  and  diminish  the  exponent  of  xf  hy  \  ;  then 
divide  the  algebraic  sum  of  these  expressions  by  the  num.ber  of  'j^  re- 
ceding co-efficients. 


CHAP.   X.J  OF   DERIVED   POLYNOMIALS.  333 

The   law   by   ^Yhich   the   co-efficients, 


X\     T, 


1.2'     1.2.3' 

are  derived  from  each  other,  is  evidently  the  same  as  that 
which  governs  the  formation  of  the  numerical  co-efficients  of 
the   terms   in   the   binomial   formula. 

The  expressions,  Y\  Z\  F,  W\  &c.,  are  called  successive  de- 
rived polynomials  of  X\  because  each  is  derived  from  the  pre- 
ceding  one   by   the   same   law   that  Y^  is   derived  from  X\ 

Generally,  any  polynomial  which  is  derived  from  another  by 
the   law  just   explained,   is   called   a  derived  polynomial. 

Recollect  that  X'  is  what  the  given  polynomial  becomes  when 
xf  is   substituted   for   x. 

Y  is   called   the  first-derived  polynomial ; 
Z^     is   called   the   second- derived  polynomial ; 

V  is   called   the   third-derived   polynomial ; 

(fee,  &c. 

We  should  also  remember  that,  if  we  make  u  =  0,  we  shall 
have  x^  —  ar,  whence  X^  will  become  the  given  polynomial,  from 
which  the  derived  polynomials  will  then  be  obtained. 

265.  Let  us  now  apply  the  above   principles  in  the  following 

EXAMPLES. 

1.  Let  it  be  required  to  find  the  derived  polynomials  of  the 
first  member  of  the   equation 

Sx^  +  6a;3  —  3:^2  _j.  2;r  +  1  =  0. 

Now,  u  being  zero,  and  x^  =  x,  we  have  from  the  law  of 
form  J  g  ^he  derived  polynomials, 

X'  =:    Sx^  +    6.t;3  -  Sx^  +  2a;  +  1  ; 
Y  z=:  12a;3  +  18:8^'  —  6x   -i-  2  ; 
Z"  -SQx^  +  mx   -6; 
V'  =  72x  +36; 
TP  =  72. 


334  ELEMENTS  OF  ALGEi^A.  [OHAP.  X 

It  should  be  remarked  that  the  exponent  of  x,  in  the  terms  1,  2, 
—  6,  36,  and  72,  is  equal  to  0;  hence,  each  of  those  terms 
disappears  in  the  following  derived  polynomial. 

2.  Let  it  be  required  to  cause  the  second  term  to  disappear 
in   the   equation 

x^  -  12^3  +  17^'2  _  9a;  +  7  =  0. 

12 
Make  (Art.  263),        a;  =  w  +  —  =1^  +  3; 

whence,  a/  =  3. 

The  transformed  equation  will  be  of  the  form  • 

and   the   operation   is   reduced   to   finding   the   values  of  the  co- 
efficients 

Yf  V^         __ 

"^'  '        2'       2.3* 

Now,   it  follows    from   the   preceding   law,   for  derived   poly- 
nomials, that 
X'    =      (3)^-12.  (3)3+ n.  (3)2-9.  (3)^-}-7,  or  X^    =-110; 
Y'    =4.(3)3-36.(3)2+34.(3)1-9,     or     -     -      Y'    =-123; 

^    =6.(3)2-36.(3)1  +  17,     or |^    =  ._    37; 

V  F 

-—  =4.f3V  — 12 —-=0. 

2.3         ^^  2.2 

Therefore,  the  transformed  equation  becomes 
^4  __  37^^2  _  i23w  -  110  =  0. 

3.  Transform  the  equation 

4^.3  _  5;^;2  _^  7^  _  9  ^  0 
into  another  equation,  the  roots  of  which  shall    exceed    those  of 
the  given  equation  by  1. 

Make,  x  z=  u  —  l;     whence     x^  =  —  1 : 

and  the  tiansformed  equation  will  be  of  the  form 


CHAP.   X.J  DERIVED   POLYNOMIALS.  335 

We  have,  from  the  principles  established, 
X'  =    4.(--l)^-    5.  (-1)2 +  7.  (-1)1- 9,  or  X'  =  -25 
F^  =12.  (-1)2 -10.  (-  1)1  +  7  .    .  ^=+29 

7^  Z' 

_=12.(-l)^-5       - 2-=-^' 

V  V 

0=    * 273=+    ^' 

Therefore,  the  transformed  equation  is, 

4u^  -  17^2  +  29?^  —  25  =  0. 

4.  What  is   the   transformed    equation,  if  the   second  term  be 
made   to   disappear   from  the   equation 

x^  -  lO;?;^  +  7a;3  +  4a;  -  9  =  0  ? 

Ans.  u^  -  33^3  _  118^2  -i52w  -  73  =  0. 

5.  What  is   the   transformed   equation,  if  the  second  term  bf» 
made   to   disappear   from   the  equation 

3a;3  +  15^2_|_25a:- 3  =  0? 

Arts.  t,3  _  1^  ^  0. 

6.  Transform   the   equation 

3:r*  -  13^3  +  7:i;2  -  8^  -  9  =  0 
into  another,  the  roots  of  which  shall   be  less  than  the  roots  of 

.       .        1 
the  given  equation  by  — . 

o 

65  84 

Ans,  3w*  —  9^3  _  4^2 ,^ _  q 

9  3 

Properties  of  Derived  Polynomials. 

266 •  We  will  now  develop  some  of  the  properties  of  derived 
polynomials. 

Let  x"^  +  Px"^-^  +  Qx^^  .  .  .  Tx-\-  U  —  0 

be  a  given  equation,  and  a,  i,  c,  c?,  &c.,  its  m  roots.     We  shall 

then  have  (Art.  248), 

sc"*  +  Pu?*^!  -f  gar'»-2  ,  .  .  —  (^  _  a)  (c  _  5)  (a;  _  c)  .  .  .  (a;  —  ^), 


836  JELEMENTS   OF   ALGEBRA.  [CHAP.    X. 

Making  x  ==  a/  -{-  u, 

or  omitting  the  accents,  and  substituting  x  -{-  u  for  x,  and  we  have 
{v  +  w)^  +  F{x  -f  u)"^^  +  .  .  .  =  {x  -{-  u  —  a)  {x  +  u  —  b)  ,  .  .; 
or,  changing  the  order  of  x  and  w,  in  the  second  member,  and 
regarding   x—  a,   x  —  b,  .  ,  ,  each  as  a  single  quantity, 


(x  -f  u)""  +F{x  -{'  w)*"-'  ...  =z{ic-\-  X  —a)  {u-\-x—b) . . .  (u-\-x—l). 
Now,    by    performing    the    operations   indicated    in    the    two 
members,  w^e  shall,  by  the   preceding  article,  obtain  for  the  first 
member, 

X  being  the  first  member  of  the  proposed  equation,  and  P",  Z,  &c., 
the  derived  polynomials  of  this  member. 

With  respect  to  the  second  member,  it  follows  from  Art.  251; 

1st.  That  the  term  involving  w^,  or  the  last  term,  is  equal  to 
the  product  (x  —  a){x  —  b^  .  .  .  (x  —  V)  of  the  factors  of  the 
proposed   equation. 

2d.  The  co-efficient  of  u  is  equal  to  the  sum  of  the  products 
of  these   m   factors,  taken  m  —  1    and   m  —  1. 

3d.  The  co-efficient  of  v?'  is  equal  to  the  sum  of  the  products 
of  these  m  factors,  taken   m  —  2   and   m  —  2 ;    and  so  on. 

Moreover,  since  the  two  members  of  the  last  equation  are 
identical,  the  co-efllcients  of  the  same  powers  of  u  in  the  two 
members  are  equal.     Hence, 

X  ^=1  (x  —  oi)  {x  —  b^  {x  —  c)  .  .  .  (x  —■  I), 
which  w^as  already  shown. 

Hence,  also,  y,  or  the  first  derived  polynomial,  is  equal  to  the 
sum  of  the  products  of  the  m  factors  of  the  first  degree  in  the  pro- 
posed equation^  taken  ra  —  1  and  m  —  1  /  or  equal  to  the  algebraic 
sum  of  all  the  quotients  that  can  be  obtained  by  dividing  X  b]/ 
each  of  the  m  factors  of  the  first  degree  in  the  proposed  equatio7i , 
ihat  is^ 

X  —-  a       X  —  b       X  —  c  X  —  I 


CHAP    X.]  EQUAL  ROOTS.  837 

Also,  — -,  that  is,  the  second  derived  polynomial,  divided  by  2, 

is  equal  to  the  sum  of   the  products  of  the  m  factors  of  the  first 
member  of  the  proposed  equation^    taken   m  —  2  and   m  —  2 ;   or 
equal  to  the  sum  of  the  quotients  obtained  by   dividing  X  by  each 
of  th&  different  factors  of  the  second  degree  ;    that  is, 
Z  X  X  X 


2         (x  —  a){x  -  b)'^  (x  —  a)  [x  —  c)  '  '  '   {x  -  h){x  -  /)' 
and   so   on.  i 

Of  Equal  Roots, 

267.  An  equation  is  said  to  contain  equal  roots,  when  its  first 
member  contains  equal  factors  of  the  first  degree  with  respect  to 
the  unknown  quantity.  When  this  is  the  case,  the  derived  poly- 
nomial, which  is  the  sum  of  the  products  of  the  m  factors  taken 
m  —  1  and  m  —  1,  contains  a  factor  in  its  different  parts,  which 
is  two  or  more  times  a  factor  of  the  first  member  of  the  pro- 
posed equation  (Art.  266) :   hence, 

There  must  be  a  common  divisor  between  the  first  member  of  the 
proposed  equation,  and  its  first  derived  polynomiaL 

It  remains  to  ascertain  the  relation  between  this  common  divi- 
sor and  the  equal  factors. 

268.  Having  given  an  equation,  it  is  required  to  discover  whether 
it  has  equal  roots,  and  to  determine   these  roots  if  possible. 

Let  us  make 

Xz^x'^^  Fx^-  +  Qx^^  +  ^  _  +  Tx+  U=0, 

and  suppose  that  the  second  member  contains  n  factors  equal  to 
X  —  a,  n^  factors  equal  to  x  —  b,    n^^  factors   equal  to  a;  —  c  .  .  ., 
and  also,  the  simple   factors   x  —p,    x  —  q,   a;  —  r  .  .  . ;  we  shall 
then  have, 
X  -  {x  —  ay  {x  —  bY  (^  —  cY'  ,  ,  ,  {x  -^p)  {x  —  q){x-'  r)     (1). 

We  have  seen  that  Y,  or  the  derived  polynomial  of  X,  is 
the  sum  of  the  quotients  obtained  by  dividing  X  by  each  of  the  m 
factors  of  the  first  degree  in  the  proposed  equation  (Art.  266). 

22 


338  ELEMENTS   OF   ALGEBRJP.  fCHAP.    X. 

Now,    since    X  contains    n    factors    equal    to   a:  --  «,   we   shall 

hav^e  n  -partial  quotients  equal  to     ;    and  the  same  reason 

irg   applies   to   each  of  the   repeated  factors,  x  —  b,  x  —  c 

Moreover,  w^e  can  form  but  one  quotient  for  each  simple  factor, 
which  is  of  the  form, 

X  X  X 

X  — ^'     X  —  q^     X  —  r  '  '  '   ' 

therefore,  the  first  derived  polynomial   is   of  the  form, 

X  —  a      X  —  o      X  —  c  X  — p      X  —  q      x  —  r 

By  examining  the  form  of  the  value  of  X  in  equation  (1), 
it   is   plain   that 

(x  -  a)«~i,       (x  -  ly--^,       {x  -  cY'-^  .  .  . 
are    factors    common    to    all    the    terms   of   the    polynomial    F; 
hence   the   product, 

(x  —  a)^-i  Y.(x  —  by-^  X  {x  —  cy-^  ... 
is   a   divisor   of    Y,     Moreover,   it   is   evident   that   it   wdll   alsc 
divide  X:   it   is   therefore   a  common  divisor  of  X  and   Y;  and 
it  is   their   greatest   common   divisor. 

For,  the   prime  factors  of  X,  are  x  —a,   x  —  b,    x  —c  .  .  .,  and 

X  —p,    X  —  q,    X  —  r  .  ,  ,  ',     now,     x  —p,    x  ^  q,    x  —  r^    cannot 

divide  JT,  since  some  one  of  them  will  be  wanting   in  some  of 

the  parts  of  P",  while  it  will  be   a  factor  of  all  the  other  parts. 

Hence,  the   greatest   common   divisor  of  X  and    F,  is 

Dzzzf^x  —  a)«-i  (x  —  6)»^-i  {x  —  cY"-^  .  .     . ;     that  is, 

The  greatest  common  divisor  is  composed  of  the  product  of  those 
factors  which  enter  two  or  more  times  in  the  given  equation^  each 
raised   to  a  power  less  by  1   than  in  the  primitive  equation. 

269.  From  the  above,  we  deduce  the  following  method  foi 
finding   the   equal    roots. 

To   discover  whether  an   equation, 

contains   any  equal   roots: 


CHAP.  IX.J  EQUAL  ROOTS.  389 

1st.  Form  Y^  or  the  derived  'polynomial  of  X ;  then  seek  for 
i}\e  greatest  common   divisor   between  X  and  Y, 

2d.  If  one  cannot  he  obtained^  the  equation  has  no  equal  roots, 
or  equal  factors. 

If  we  find  a  common  divisor  D,  and  it  is  of  the  first  degree, 
or  of  the   form   a*  —  A,  make  x  —  h  =  0,  whence  x  =  h. 

We  then  conclude^  that  the  equation  has  two  roots  equal  to  li, 
and  has  but  one  species  of  equal  roots,  from  which  it  may  be 
freed   by   dividing   X  by    (x  ■—  h)^. 

If  D  is  of  the  second  degree  with  reference  to  or,  solve  tht 
equation  D  z=  0.  There  may  be  two  cases ;  the  two  roots  will 
be   equal,  or   they  will   be   unequal. 

1st.  When  we  find  D  =  (x  —  h)^,  the  equation  has  three  roots 
equal  to  h,  and  has  but  one  species  of  equal  roots,  from  which 
it   can   be   freed   by  dividing  X  by  (x  —  h)^. 

2d.  When  D  is  of  the  form  (x  —  h)  {x  —  h')^  the  proposed 
equation  has  tioo  roo'ts  equal  to  h,  and  two  equal  to  h',  from 
which  it  may  be  freed  by  dividing  JT  by  {x  —  hy  {x  —  h')'^, 
or  by  i)2^ 

Suppose  now  that  D  is  of  any  degree  whatever  ;  it  is  necessa7y^ 
in  order  to  know  the  species  of  equal  roots,  and  the  number 
of  roots  of  each   species,  to   solve   completely    the  equation, 

D  =  0. 

Then,  every  simple  root  of  the  equation  D  =  0  will  be  twice  a 
root  of  the  given  equation;  every  double  root  of  the  equation  D  =  0 
will  be  three  times  a  root  of  the  given  equation  '  and  so  on. 

As   to   the   simple   roots   of 

X=0, 

we  begin  by  freeing  this  equation  of  the  equal  factors  contained 
in  it,  and  the  resulting  equation,  JT'  =  0,  will  make  known  the 
simple  roots. 


340  ELEMENTS   OF  ALGEBRA.   *  [CHAP.    X. 


EXAMPLES. 


1.  Determine   whether   the  equation, 

2x^  —  12a;3  +  19a;2  —  6a;  +  9  =  0, 

eontains   equal   roots. 

We   nave   for   the  first  derived   polynomial, 

Sx^  ~  36a;2  +  38a;  —  6. 

Now,  seeking  for  the  greatest  common  divisor  of  these  poly- 
nomials, we  find 

D  z=  X  —  2  =z  0,      whence      x  =zS: 
hence,  the  given   equation   has  two   roots   equal   to   3. 
Dividing  its   first  member  by   {x  —  S)\  we   obtain 

2a;2  +  1  =  0  ;     whence,     x  =  ±:  -—V—  2. 

The  equation,  therefore,  is  completely  solved,  and  its  roots  are 

3,     3,      +1/Z:2~and     ~  y/^=2. 


2V  2' 

2.  For   a  second   example,  take 

x5  —  2x^  +  Sx^  —  7ii;2  +  8a;  —  3  =  0. 
The  first  derived  polynomial  is 

5a;*  -  8a;3  +  9a;2  -  14a;  +  8  ; 
and  the  common  divisor, 

a;2  -  2a;  +  1  =  (a:  -  1)2  : 
hence,  the  proposed  equation  has  three  roots   equal  to  1. 
Dividing  its  first  member  by 

{x  -  1)3  =  a;3  -  3a;2  +  3a;  -  1, 
tlie   quotient  is 

a;2  +  a;  +  3  =^  0  ;     whence,     x  = ^ ; 

thus,  the  equation  is  completely  solved. 


CHAP.  X.J  EQUAL  BOOTS.  841 

3.  For  a  third  example,  tako.  the   equation 

x^  +  5a;6  +  6x^  -  6x^  -  I5x^  —  Sx^  +  8a;  -f-  4  =  0. 
The  first  derived  polynomial  is 

7x^  +  SOx^  +  30a;*  —  24x^  —  45a:2  _  6a;  +  8  ; 
and   the   common   divisor  is 

x^  +  3a;3  +  a;2  —  3a;  —  2. 
The   equation, 

x^  +  ^x^  +  x^  -^Sx  —  2z=0, 
cannot  be  solved  directly,  but  by  applying  the  method  of  equal 
roots   to   it,  that  is,  by  seeking  for   a   common  divisor  between 
its  first   member   and   its   derived   polynomial, 

4a;3  +  9x^  +  2a;  —  3  : 
we  find  a  common  divisor,  a;  +  1  ;  which  proves  that  the  square 
of  a;  +  1   is  a  factor  of 

x^  +  3a;3  +  a;2  —  3a;  —  2, 
and   the   cube    of  a;  +  1,    a    factor   of  the   first   member   of    the 
given    equation. 
Dividing 

X-  +  3a;3  +  a;2  —  3a;  —  2     by  {x  +  1)2  =  a;2  +  2a;  +  1, 
we   have    a;2  +  a;  —  2,   which   being   placed  equal   to   zero,  gives 
the   two   roots   a;  =  1,   a;  =  —  2,  or   the   two  factors,    x  —  I    and 
a;  +  2.     Hence,  we   have 

x^  +  3a;3  H-  a;2  -  3a;  -  2  =  (a;  +  1)^  (^  -  1)  (a;  +  2). 
Therefore,  the  first  member  of  the  proposed  equation  is  equal  to 
{x  +  1)3  {x  -  1)2  (a;  +  2)2 ; 
that  is,  the  proposed  equation  has  three  roots  equal  to   —•  1,  two 
equal   to    +1,  and   two   equal   to    —  2. 

4.  What  is  the  product   of  the   equal  factors  of  the  equation 
x'  —  7a;6  +  10a;5  +  22a;*  -  43a;3  —  35a;2  +  48a;  +  36  =  0  ? 

Ans.    (a;  — 2)2  (a; -3)2  (a;  4- 1)3. 

5.  What  is   the  product  of  the  equal  factors  in  the  equation, 

x^  -  3a;«  +  9x^  -  19a;*  +  27a;3  -  33a;2  +  27a;  -  9  =  0  ? 

Ans,  (x  —  ^Y(x'^^\-^Y, 


342  ELEMElsTS   OF   ALGEBRA.  [CHAP.   X 

Elimination, 

270»  We  have  already  explained  the  methods  of  eliminating 
Ohe  unknown  quantity  from  two  equations,  when  these  equations 
ure  of  the  first  degree  with  respect  to  the  unknown  quantities. 

When  the  equations  are  of  a  higher  degree  than  the  first, 
tlie  methods  explained  are  not  in  general  applicable.  In  this 
case,  the  method  of  the  greatest  common  divisor  is  considered  the 
best,  and   it   is  this  method  that  we  now  propose  to  investigate. 

One  quantity  is  said  to  be  a  function  of  another  when  it  de- 
pends upon  that  other  for  its  value ;  that  is,  when  the  quan- 
tities are  so  connected,  that  the  value  of  the  latter  cannot  be 
changed  without  producing  a  corresponding  change  in  the  former. 

27  !•  If  two  equations,  containing  two  unknown  quantities,  be 
combined,  so  as  to  produce  a  single  equation  containing  but  one 
unknown  quantity,  the  resulting  equation  is  called  a  final  equa- 
tion ;  and  the  roots  of  this  equation  are  called  compatible 
values  of  the  unknown  quantity  which  enters  it. 

Let  us   assume   the   equations, 

P  ^  0     and     §  ==  0, 

in  which  P  and  Q  are  functions  of  x  and  y  of  any  degree 
whatever ;  it  is  required  to  cc^mbine  these  equations  in  such  a 
manner   as   to   eliminate   one   of  the   unknown   quantities. 

If  we  suppose  the  final  equation  involving  y  to  be  found,  and 
that  y  =:  a  is  a  root  of  this  equation,  it  is  plain  that  this  value 
of  y,  in  connection  with  some  value  of  ir,  will  satisfy  both 
equations. 

If  then,  we  substitute  this  value  of  y  in  both  equations,  there 
Mill  result  two  equations  containing  only  x^  and  these  equations 
will  have  at  least  one  root  in  common,  and  consequently,  their 
first  members  will  have  a  common  diviscr  involving  x  (Art.  246), 

This  common  divisor  will  be  of  the  first,  or  of  a  higher  degree 
with  respect  to  ic,  according  as  the  particular  value  of  y  z=z  a  cor 
responds  to  one  or  more  values  of  x. 


CHAP.   XI.j  ELIMINATION.  343 

Conversely,  every  value  of  y  which,  being  substituted  in  the 
two  equations,  gives  a  common  divisor  involving  x,  is  necessarily 
a  compatible  value,  for  it  then  satisfies  the  two  equations  at  the 
same  timxe  with  the  value  or  values  of  x  found  from  this  common 
divisor  when  put  equal  to  0. 

272.  We  will  remark,  that,  before  the  substitution,  the  first 
members  of  the  equations  cannot,  in  general,  have  a  common  divi- 
sor which  is  a  function  of  one  or  both  of  the  unknow^i  quantities. 

For,  let   us   suppose,   for   a   moment,   that   the  equations 

P  =  0     and     Q  —  0, 
are  of  the   form 

P^  X  i?  =  0     and     6^  X  i2  =  0, 
R  being   a   function  of  both   x   and   y. 

Placing  R  z=zO,  we  obtain  a  single  equation  involving  two 
unknown  quantities,  which  can  be  satisfied  w4th  an  infinite  number 
of  systems  of  values.  Moreover,  every  system  which  renders  R 
equal  to  0,  would  at  the  same  time  cause  P' .  R  and  Q'  ,R  to 
become  0,  and  consequently,  would  satisfy  the  equations 
P  =  0     and     §  =  0. 

Thus,  the  hypothesis  of  a  common  divisor  of  the  two  poly, 
nomials  P  and  Q,  containing  x  and  y,  brings  with  it,  as  a  con- 
sequence, that  the  proposed  equations  are  indeterminate.  There- 
fore, if  there  exists  a  common  divisor,  involving  x  and  y,  of  the 
two  polynomials  P  and  Q,  the  proposed  equations  will  be  inde- 
terminate,  that  is,  they  may  be  satisfied  by  an  infinite  number 
of  systems  of  values  of  x  and  y.  Then  there  is  no  data  to 
determine  a  final  equation  in  y,  since  the  number  of  values  of  y 
is   infinite. 

Again,  let   us  suppose  that  P  is  a  function  of  x  only. 

Placing  R  =:  0,  we  shall,  if  the  equation  be  solved  with 
reference  to  x,  obtain  one  or  more  values  for  this  unknown 
quantity. 

Each  of  these  values,  substituted  in  the  equations 
P\R=:0    and     g^  i2  =  0, 


344  ELEMENTS   OF  ALGEBRA*  [CHAP.   X 

will  satisfy  them,  whatever  value  we  may  attribite  to  y,  sinco 
these  values  of  x  would  reduce  R  to  0,  independently  of  y. 
Therefore,  in  this  case,  the  proposed  equations  admit  of  a  finite 
number  of  values  for  x^  but  of  an  infinite  number  of  values  for 
y  and  then,  therefore,  there  cannot  exist  a  final  equation  in  y. 
Hence,  when  the  equations 

are  determinate,  that  is,  when  they  admit  only  of  a  limited 
number  of  systems  of  values  for  x  and  y,  their  first  members 
cannot  have  for  a  common  divisor  a  function  of  these  unknow:i 
quantities^  unless  a  particular  substitution  has  been  made  for  one 
of  these  quantities. 

273»  From  this  it  is  easy  to  deduce  a  process  for  obtaining 
the  final  equation  involving  y. 

Since  the  characteristic  property  of  every  compatible  value 
of  y  is,  that  being  substituted  in  the  first  members  of  the  two 
equations,  it  gives  them  a  common  divisor  involving  .r,  which 
they  had  not  before,  it  follows,  that  if  to  the  two  proposed 
polynomials,  arranged  with  reference  to  x^  we  apply  the  process 
for  finding  the  greatest  common  divisor,  we  shall  generally  not 
find  one.  But,  by  continuing  the  operation  properly,  we  shall 
arrive  at  a  remainder  independent  of  ar,  but  which  is  a  function 
of  y,  and  which,  placed  equal  to  0,  will  give  the  required  final 
equation. 

For,  every  value  of  y  found  from  this  equation,  reduces  to 
rero  the  last  remainder  in  the  operation  for  fmding  the  common 
divisor ;  it  is,  then,  such  that  being  substituted  in  the  preceding 
remainder,  it  will  render  this  remainder  a  common  divisor  of  the 
first  members  P  and  Q.  Therefore,  each  of  the  roots  of  the 
equation  thus  formed,  is  a  compatible  value  of  y, 

274.  Admitting  that  the  final  equation  may  be  completely 
solved,  which  would  give  all  the  compatible  values,  it  would 
afterward  be  necessary  to  obtain  the  corresponding  values  of  x. 
Now,  it  is  evident  that  it  would  be  sufficient  for  this,  to  sub- 
stitute the  different  values  of  y  in  the  remainder  preceding  the 


CHAP.   X.]  KLIMINATIOIS^.  845 

last,  put  the  polynomial  involving  x  which  results  from  it,  equal 
to  0,  and  find  from  it  the  values  of  x\  for  these  polynomials 
are  nothing  more  than  the  divisors  involving  re,  which  become 
common  to  A  and  B. 

But  as  the  final  equation  is  generally  of  a  degree  superior  to 
the  second,  we  cannot  here  explain  the  methods  of  finding  the 
values  of  y.  Indeed,  our  design  was  principally  to  show  that, 
two  equations  of  any  degree  being  given,  we  can,  without  supposing 
the  resolution  of  any  equation^  arrive  at  another  equation,  contain- 
ing only  one  of  the  unknown  quantities  which  enter  into  the  pro- 
posed   equations, 

EXAMPLES. 

1.  Having  given  the  equations 

x^  +  xy  -\-y^  --    1=0, 
a;3  +  y3  =  0, 
to  find  the  final  equation  in  y. 

First  Operation, 

a;2  _j.  ^y  ^  y2  _  1 


x^  +  y^ 

x^  -\-  yx'^  +  (?/2  —  X)x 


X  — y 


—  yx'^  —  (y3__l)^_^y3 

—  yx'^—y'^x  —  y^     +y 


,  a:  +  2y3  — -  y  =  1st  remainder. 

Second  Operation, 

x^ -i-   yx  +2/2_i  \\x  +  2y3  —  y 

x^+{2y^-y)x  ^^x^{2y^^2y) 

-{2y^-2y)x+    y^-^1 
—  (2y3  —  2y)  x  -  4y^  +  ^y*  —  2y2  * 
4y6  —  6y^  +  3y2  —  1. 
Hence,  the  final  equation  in  y,  ,is 

4y«  ~  6y^  +  3y2  —  1  =  0. 


84:6  ELEMENTS   OF  ALGEBRA.  [CKAP,   X. 

If  it  were  required  to  find  the  final  equation  in  u?,  we  observe 
that  X  and  y  enter  into  the  primitive  equations  under  the  same 
forms ;  hence,  x  may  be  changed  into  y  and  y  into  x,  without 
destroying  the  equality  of  the  members.     Therefore, 

4x^  —  6x^  +  3a:2  —  1  _o 
1*3  the  final  equation  in  x, 

2.  Find  the  final   equation   in  y,  from  the  equations 
x^-St/x^  +  (33/2  -  y  +  1)  ^  --  y3  ^  2^.2  _2y  =  0, 
x^  —  2yx  +      y"^  —  y  =:0. 

First  Operation, 
x^  —  Syx^  +  (3?/2  __  y  4-  1)  a;  —  2/3  _|_  ^2  _  2y||a;2  —  2xy  +  y^-^^ 
x^  —  2yx'^  +    {y^  —  y)x  X   —y 

—  yx^  +  (23/2  -f  1)  a;  —  2/3  -f  y2  _  2y 

—  yx'^  +    22/2^;  —  2/^  +  y^ 


ar  — 2?/ 
Second  Operation, 

«2  —  22:?/  4-  y^  —  y 
a;2  —  22"y 


•2y 


2/^-y. 

Hence,  y"^  —  y  =z  0^ 

is   the  final   equation   in   y.     This   equation  gives 
y  =  \     and    y  =  0. 
Placing   the   preceding   remainder   equal    to   zero,    and   substW 
tuting   therein   t^e  values  of  y, 

y  =  1     and     y  =  0, 
we  find  for  the    corresponding  values  of 'a?, 
a;  =  2     and     a;  ==  0  ; 
from  which  <t^    ^iven  eqic.ations  may  be   entirely  solved. 


CHAPTER  XI. 

SOLUTION     OF     NC'MER.OAL     EQUATIONS     CONTAINING     BUT      ONE     UNKNOWN 

QUANTITY. — Sturm's  theorem. — cardan's  rule. — horner's  method. 

275.  The  principles  established  in  the  preceding  chapter,  are 
applicable  to  all  equations,  whether  the  co-efficients  are  numerical 
or  algebraic.  These  principles  are  the  elements  which  are  em- 
ployed in  the  solution  of  all  equations  of  higher  degrees. 

Algebraists  have  hitherto  been  unable  to  solve  equations  of  a 
higher  degree  than  the  fourth.  The  formulas  which  have  been 
deduced  for  the  solution  of  algebraic  equations  of  the  higher 
degrees,  are  so  complicated  and  inconvenient,  even  when  they 
can  be  applied,  that  we  may  regard  the  general  solution  of  an 
algebraic  equation,  of  any  degree  whatever,  as  a  problem  more 
curious   than   useful. 

Methods  have,  however,  been  found  for  determinmg,  to  any 
degree  of  exactness,  the  values  of  the  roots  of  all  numerical 
equations ;  that  is,  of  those  equations  which,  besides  the  unknown 
quantity,  involve  only  numbers. 

It  is  proposed  to  develop  these  methods  in  this  chapter. 

276.  To  render  the  reasoning  general,  we  will  take  the 
equation, 

X=zx'^  +  Fx"^^  +  Qx"^^  +  .  .  .   V  =  i}. 
in  which  P,   Q  .  .  ,  denote   particular   numbers  which   are   real, 
and    either  positive   or   negative. 

if  we  substitute  for  x  a  number  a,  and  denote  by  A  what 
A"  becomes  under  this  supposition ;  and  again  substitute  a  -{-  zl 
ioi  X.  and  denote  the  new  polynomial  by  A^ :  then^u  may  be 
taken  i:j  small,  that  the  difference  between  A'  and  A  shall  be 
less   than   any   assignable   quantity. 


348  ELEMENTS   OF  ALGEBRA.  [CHAP.   XL 

If,  now,  we  denote  hj  B,   0,  D,  ,  .  ,  .  what  the   co-efficients 

Z       V 
F,    — ,    - — -    (Art.  264),  become,  wh«n   we    make    x  =  a,   we 

shall    have, 

A' =  A +Bu+  Cu'^  +  Du^+  .  .  .   +u'^    ...     (1); 
whence, 

A'  —  A  =  Bu+Cu'^  +  Du^  +  .  .  .  +  2^«     .    .    .     (2). 

It  is  now  required  to  show  that  this  difference  may  be  ren- 
dered  less  than  any  assignable  quantity,  by  attributing  a  value 
sufficiently  small  to  u. 

If  it  be  required  to  make  the  difference  Isotween  A^  and  A 
less  than  the  number  iV,  we  must  assign  a  value  to  u  which 
will  satisfy  the  inequality 

Bu  +  Cu^  +  Du^  + W^^JSr    -    .    -    (3). 

Let  us  take  the  most  unfavorable  case  that  can  occur,  viz., 
let  us'  suppose  that  every  co-efficient  is  positive,  and  that  each 
is  equal  to  the  largest,  which  we  will  designate  by  J^,  Then 
any  value  of  u  which  will  satisfy  the  inequality 

K{u  +  u^  +  u^-\- W^XJSr     .    -    -      (4), 

will  evidently  satisfy  inequality  (3). 

Now,  the  expression  within  the  parenthesis  is  a  geometrical 
progression,  whose  first  term  is  u,  whose  last  term  is  u^,  and 
whose  ratio  is  u ;    hence  (Art.   188), 

W  +  I                                  TO  + 1 
U-{-  U^+  U^+  ,  .  .U"^  = = r=- X    (1  —  W^). 

u    —    I        1  —-  u         1  —  u       ^  ' 

Substituting  this  value  in  inequality  (4),  we  have, 
Ku 


1  -t^ 


(1  -  w'^X  ilT    -    .    .    .     (5). 


N 
\{  now  we  make  u  =  -— -,  the  first  factor  of  the  first  mem 

N  ^  K' 

^         I    ii  less 

than   1,  the  second  factor  is  less  than  1 ;  hence,  the  fijst  mena 
ber  is  less  than  N, 


CHAP.   XU  NUMERICAL  EQUATIONS.  349 

We  conclude,  therefore,  that  u  z=  — - — -:,    and    every  smaller 

value  of  u,  will  satisfy  the  inequalities  (3)  and  (4),  and  conse- 
quently, make  the  difference  between  A'  and  A  less  than  any 
assignable  number  JV. 

If  in  the  value   of  A\  equatkn  (1),  we  make  u^=^  ,  it 

Is  plain  that  the  sum  of  the  terms 

Bu  +  Cv?  +  Bu^  +  .  .  .  -w*" 
will  be  less  than  A^  from   what   has  just  been  proved ;    whence 
we  conclude  that 

In  a  series  of  terms  arranged  according  to  the  ascending  powers 
of  an  arbitrary  quantity^  a  value  may  he  assigned  to  that 
so  small,  as  to   make  the  first  term    numerically  greater  than  the 
sum  of  all  the  other  terms. 

First  Principle. 

277«  If  t'^o  numbers  p  and  q,  substituted  in  succession  in  the 
place  of  X  in  the  first  member  of  a  numerical  equation,  give  results 
affected  with  contrary  signs,  the  proposed  equation  has  a  real  root, 
comprehended  between  these  two  numbers. 

Let  us  suppose  that  p,  when  substituted  for  x  in  the  first 
member  of  the  equation 

X  =  0,      gives      +  B, 
and  that  q,  substituted  in  the  first  member  of  the  equation 
X  =  0,      gives      —  jK'. 

Let  us  now  suppose  x  to  vary  between  the  values  of  p  and  q 
by  so  small  a  quantity,  that  the  difference  between  any  two 
corresponding  consecutive  values  of  X  shall  be  less  than  any 
assignable  quantity  (Art.  276),  in  which  case,  we  say  that  X  is 
subject  to  the  law  of  continuity,  or  that  it  passes  through  all 
the  intermediate  values  between  H  and    —  Ii\ 

Now,  a  quantity  which  is  constantly  finite,  and  subject  to  the 
'aw  of  continuity,  cannot  change  its  sign  frcm  positive  to  nega 


850  ELEMENTS   OF   ALGEBRA.  [CHAP.   XI. 

tive,  or  from  negative  to  positive,  without  passing  through  zero : 
hence,  there  is  at  least  one  number  between  'p  and  q  which  will 
satisfy    the   equation 

jr=o, 

and  consequently,  one  root  of  the  equation  lies  between  these 
numbers. 

278.  We  have  shown  in  the  last  article,  that  if  two  numbers 
be  substituted,  in  succession,  for  the  unknown  quantity  in  any 
equation,  and  give  results  affected  with  contrary  signs,  that  there 
will  be  at  least  one  real  root  comprehended  between  them.  We 
are  not,  however,  to  conclude  that  there  may  not  be  more  than 
one;  nor  are  we  to  infer  the  converse  of  the  proposition,  viz., 
that  the  substitution,  in  succession,  of  two  numbers  which  include 
roots  of  the  equation,  will  necessarily  give  results  affected  with 
contrary   signs. 

Second  Principle » 

279.  When  an  uneven  number  of  the  real  roots  of  an  equation 
is  comprehended  between  two  numbers^  the  results  obtained  by  sub- 
stituting  these  numbers  in  succession  for  x  in  the  first  member^  will 
have  contrary  signs ;  but  if  they  comprehend  an  even  number  of 
roots^  the  results  obtained  by  their  substitution  will  have  the  same  sign. 

To  make  this  proposition  as  clear  as  possible,  denote  by 
a,  b,  c,  .  ,  .  those  roots  of  the  proposed  equation, 

X=:0, 

which  are  supposed  to  be  comprehended  between  p  and  q,  and 
by  Y,  the  product  of  the  factors  of  the  first  degree,  with  refer- 
ence to  X,  corresponding  to  the  remaining  roots  of  the  given 
equation. 

The  first  member,  X,  can   then   be   put   under   the   form 

{x  ~  a){x  —  b){x  —  c)  .  .  .    X  T=zO. 
Now,  substituting  p  and   q  in   place   of  x,  in   the   first   mem- 
Der,  "Te   shall   obtain    the   two   results, 

(p-a){p-b){p-c)  .  .  ,    X  Y', 
iq-a){q-b)(q--)  .  .       X   Y". 


CHAP.   XI.]  NUMERICAL   EQUATIONS.  351 

JP  and  Y^^  representing  what  Y  becomes,  when  we  replace  in 
succession,  x  by  p  and  q.  These  two  quantities  Y^  and  ]P^,  are 
affected  w^ith  the  same  sign ;  for,  if  they  were  not,  by  the  first 
principle  there  would  be  at  least  one  other  real  root  com. 
prised  between  p  and  g,  which  is  contrary  to  the  hypothesis. 
To  determine  the  signs  of  the  above  results  more  easily, 
divide   the  first  by  the   second,  and  we   obtain 

{p  —  a)[p—h){p  —  c)  ,  ,  .    X  Y' 
{q-a){q-b){q-c)  .  ,  .    XY^' 
which   can   be  written   thus, 

p  —  a       p  —  b       p  —  c  Y' 

q  —  a        q  —0        q  —  c  Y^^ 

Now,  since   the   root  a   is   comprised  between  p   and   g,  that 
is,    is    greater   than   one   and    less    than   the    other,   p  —  a    and 

q  —  a  must   have   contrary  signs  ;    also,  p  —  h    and    q  —  h  must 
have   contrary  signs,  and   so   on. 
Hence,  the   quotients 

p  —  a     p  —  h     p  —  c 

,     J, ,    &c., 

q  —  a      q  —  b      q  —  c 

are   all   negative. 

Moreover,  •  -—y     is  essentially  positive,  since   Y'  and   Y''  are 

affected  w^ith   the   same   sign ;   therefore,  the   product 
p  —  a      p  —  h      p  —  c  Y' 

X  7     X  X        •      •      •  -xy-n^ 

q  —  a       q  —  b       q  —  c  Y 

will  be  negative^  when  the  number  of  roots,  a,  5,  c  .  .  .,  com 
prehended  between  p  and  q^  is  uneven,  and  positive  when  the 
number  is   even. 

Consequently,  the   two   results, 

{p-a){p-b){p  ^c)  .  ,  ,    X  Y', 
and  {<l-<^){q  -'h)[q  —  c)  .  .  ,    X  Y'. 

will  have  contrary  signs  w^hen  the  number  of  roots  comprised 
between  p  and  q  is  uneven,  and  the  same  sign  when  the  num- 
ber is  even 


S52  ELEMENTS   OF   ALGEBR^.  [CHAP.   XL 

Third  Principle. 

280.  If  the  signs  of  the  alternate  terms  of  an  equation  be 
changed^   the   signs  of  the  roots  will  he  changed,  ^ 

Take   the   equation, 

^m  +  p^m-i  _|_  ^^^2  .  .  .   +  cr  —  0     -  -     (1)  ; 

and   by  changing   the   signs  of  the   alternate   terms,  we  have 

x^  —  Px"^^  +  Qx"^'^  .  ,  .   ±U=0     -  -     (2), 

or,  -  ic"*  +  Px"^^  —  Qx'^'^  .  ,  .  zf  U=iO     -  -     (3). 

But  equations  (2)  and  (3)  are  the  same,  since  the  sum  of  the 
positive  terms  of  the  one  is  equal  to  the  sum  of  the  negative 
terms  of  the   other,  whatever  be  the  value  of  x. 

Suppose  a  to  be  a  root  of  equation  (1)  ;  then,  the  substitution 
of  a  for  X  will  verify  that  equation.  But  the  substitution  of 
—  a  for  iT,  in  either  equations  (2)  or  (3),  will  give  the  same 
result  as  the  substitution  of  +  a,  in  equation  (1)  :  hence  —  a, 
is   a   root   of  equation  (2),  or  of  equation  (3). 

We  may  also  concl^ide,  that  if  the  signs  of  all  the  terms 
be   changed,  the   signs   of  the  roots   will   not   be   altered. 

Limits  of  Real  Roots, 

281.  The  different  methods  for  resolving  numerical  equations, 
consist,  generally,  in  substituting  particular  numbers  in  the  pro- 
posed  equation,  in  order  to  discover  if  these  numbers  verify  it, 
or  whether  there  are  roots  comprised  between  them.  But  by 
reflecting  a  little  on  the  composition  of  the  first  member  of 
the  general   equation, 

Xm  +  p^rn^l  +   Qx'^'2'  ,    ,    ,    ^  Tx  +    TJ  —  0, 

we  become  sensible,  that  there  are  certain  numbers,  above  which 
it  would  be  useless  to  substitute,  because  all  numbers  above  a 
certain   limit  would  give   positive   results. 


CHAP.   XI.i  LIMITS  OF  REAL    ROOTS.  853 

282.  It  is  now  required  to  determine  a  number^  which  being 
substituted  for  x  in  the  general  equation^  will  render  the  first  term 
X™  greater  than  the  arithmetical  sum  of  all  the  other  terms  ; 
tliat  is,  it  is  required  to  find  a  number  for  x  which  will  render 

J^et  k  denote  the  greatest  numerical  co-efficient,  and  substitute 
it  in  place  of  each  of  the  co-efficients;  the  inequality  will  then 
become 

x"^  >  kx"^^  +  Au'^-2  -{.,,,   J^Jcx  +  k, 

It  is  evident  that  everj  number  substituted  for  x  which  will 
satisfy  this  condition,  will  satisfy  the  preceding  one.  Now, 
dividing   both  members  of  this  inequality  by  x^^  it  becomes 

i>A  +  A  + A+  . .  +_A_  + A. 

^  X  ^  x'^^  x^^         ^  x^-^  ^  x^ 

Making  x  =  k,  the  second  member  reduces  to  1  plus  the 
sum  of  several  fractions.  The  number  k  will  not  therefore 
satisfy  the  inequality;  but  if  we  make  x  =  k -{- 1,  we  obtain 
for   the   second  member   the   expression, 

A?  A/  A7  fC  ,  fC 


^^  +  1  ^  (^  +  1)2    '^  (^  +  1)3    '     "     '    (^  +  1)^1  ^    (Ic  +  1)"*' 

Tliis  is   a  geometrical  progression,  the  first  term  of  which  is 
the  last  term,     ..,    .   ^.    ,    and  the  ratio,    ~ — --— :    hence, 

'      I U  _J_  1  \m'  '      Z»  _i_   1  '  ' 


the  expression  reduces  to  / 

k k 

1  .  "■  (^  +  1)-' 


k-\-  1 
which  is  evidently  less  than  1...  ...^ixiom  ri^od  §nlbiyia 

Now,  any  number  .jI>  (^  +  l),jufein  .plai&s  of  ^wi^  render 

the  sum  of  the  fractions 1 +  .  .  .  still   l^ss  :   ferefore,  _^ 

X         ^'-^  gnigoqqua.  ^d  57/0  W 

nuhsUtuteE  for  x,  we7r  render  the  first   term   x"*  greater  ihau   the 
arithmetical  sum  of  aJ^  tke^mr'teM     '"'^  =  "^  '^^^'8  ''""''' 

23 


854  ELEMENTS   OF   ALGEBRA.  [CHAP.   XI. 

283.  Every  number  which  exceeds  the  greatest  of  the  positive 
foots  of  an  equation,  is  called  a  superior  limit  of  the  positive  roots. 

From  this  definition,  it  follows,  that  this  limit  is  susceptible 
,)f  an  infinite  number  of  values.  For,  when  a  number  is  found 
to  exceed  the  greatest  positive  root,  every  number  greater  than 
this,  is  also  a  superior  limit.  The  term,  however,  is  generally 
applied  to  that  value  nearest  the  value  of  the  root. 

Since  the  greatest  of  the  positive  roots  will,  when  substituted 
for  a?,  merely  reduce  the  first  member  to  zero,  it  follows,  that 
we  shall  be  sure  of  obtaining  a  superior  limit  of  the  positive 
roots  by  finding  a  number,  which  substituted  in  place  of  x,  renders 
the  first  member  positive,  and  which  at  the  same  time  is  such,  that 
every  greater    number   will  also  give  a  positive   result;   hence. 

The  greatest  co-efficient  of  x  plus  1,  is  a  superior  limit  of 
the  positive   roots » 

Ordinary  Limit  of  the  Positive  Boots, 

284,  The  limit  of  the  positive  roots  obtained  in  the  last  article, 
is  commonly  much  too  great,  because,  in  general,  the  equation 
contains  several  positive  terms.  We  will,  therefore,  seek  for  a 
limit  suitable  to  all  equations. 

Let  x^-^  denote  that  power  of  x  that  enters  tHe  first  nega- 
tive term  which  follows  x^,  and  let  us  consider  the  most  unfavor- 
able case,  viz.,  that  in  which  all  the  succeeding  terms  are  negative, 
and  the  co-efficient  of  each  is  equal  to  the  greatest  of  the  nega- 
tive co-efficients  in  the  equation. 

Let  S  denote  this  co-efficient.     What  conditions  will  render 

^m  y  gg^m^  J^  Sx"^-^"^  +  ,  ,  ,  Sx  +  S  1 

Dividing  both  members  of  this  inequality  by  x*^,  we  hav» 

Now,  by  supposing 

X  =  \/ S+  1,     or  for  simplicity,  making   'wS^  S\ 
which  gives,  S  =  S^^,    and    x  =  S' +  I, 


CHAP.   XI.]  LIMITS   OF  POSITIVE   ROOTS.  855 

the  second  member  of  the  inequality  will  become, 


which  is    a  geometrical   progression,   of  which  '/  o/  ,  ■.  \^   i^   the 

first  term,  and  the  ratio.     Hence,  the  expression   for   the 

o  +  1 

sum  of  all  the  terms  is  (Art.  188),  ' 

S'^ S'^ 


^^+1 


- 1 


{S'+iy-'^    {S'+ 1)" 


Moreover,  every  Yiumber  >  ^^  +  1  or  'l/~S'+  1,   will,   when 
substituted  for  «,  render  the  sum  of  the  fractions 
S  S 

still  smaller,  since   the   numerators   remain  the   same,  while  the 
denominators  are  increased.      Hence,  this  sum  will  also  be  less. 

Hence,  ^  S  +  1,  and  every  greater  number,  being  substituted 
for  X,  will  render  the  first  term  x^  greater  than  the  arithmetical 
sum  of  all  the  negative  terms  of  the  equation,  and  will  conse 
quently  give  a  positive  result  for  the  first  member.     Therefore, 

Thai  root  of  the  numerical  value  of  the  greatest  negative  co-effi- 
cient whose  index  is  equal  to  the  number  of  terms  which  precede 
the  first  negative  term^  increased  hy  1,  is  a  superior  limit  cf  the 
positive  roots  of  the  equation.  If  the  coefficient  of  a  term  is  0, 
/Ae  term  must  still  be  counted. 

Make  n  =  1,  in  which  case  the  first  negative  term  is  the 
gecond  term  of  the  equation ;   the  limit  becomes 

2/^+1=^+1; 
that  is,  the  greatest  negative  co-efficient  plus  1. 

Let  w  =  2  ;  then,  the  limit  is  '^^^+  1.  When  n  =  3.  the 
limit  is  y  S-{-  1. 


EXAMPLES, 

equation      ^ 

/        1  --}-  a 

2.  What  is   the   superior  lii(8Sro|,Al^gipggi(ft33ee<0oIfe  *rf  tl»«~ 
equation 

3.  What  is   the   superior  limit  of   the    positive  roots   of  the 
^ii^f^JiioijIIiY/    J  4."^gr\r  10  1  +  "^^  <  lacJrnun   ^lavs  ^isyooioM 

giioffoillfiyitf  %^;?r-GeT55lftf^i  jtsi  lo*!  beiifibadus 
In  this  example,  we  see  that?jthe  sdaond  term  is  wanting,  that 
is,  its  co-efficient  is  zero  ;*  btt^tei  teW  must  still  be  counted  in 
^'S?%Iik^';oai«''Cd?-  iiiMpyi  4?»liS&^iftatefl!)h%J^rcetl){M«^tiiid3 

X  IS  zero.     Hence, 

bajirjiigcfjja  gnisd  ^lodirmxi.  "lat^o'ig  X'^^Z^   ^-^^^  f^  +  ^  \/  {Q')^©!! 

iBbiJomflJi'iE  Oil  J  riBrlV 'lolijdig  "^  ^aiW  J&'iftJvOflj  loBxroi  Wm  ^s-.  -idi 
^^^jintfe^^f^e^ife  i5^idk^]l.§a^lJwlw»l@i-!iMihbgflBgiat  o^ill^^deAdiW-^ 
^*^\^dti%iM]!>ndii^gii!fefrr  dairl  orl)  lol  iluao'i  ovijiaoq  b  9Yi§  x^^«^"P 


v,x  ..  ^.u.  X,  i^^a   superior  iimirot   the  positive  roots.     In   tha' 
last  article  we  lound  a  limit  still  less;  and  we  now  propose  to 
fisditfee  ifiMSlle^iJl^lt^  fif'^^Krf^  iM?befe^''^'^   ^^   ,1  -  *«   sieM 

Let  A"=0 

DC  the  proposed  equation^  ^  llf  ^  lis  tquatJf)n  we  make  x^y/\.vL 
^  being  aAitrarj,  wJ  sl^  o%!g?l'(irl!'Mf,  ^''^^''"^^'^'  "^'^  *'^'  ^"""^'^ 

^'+^'^  +  2"^^  •••+«"  =  Or  f^-.i,i^i, 


a  number  for  a/,  whic\$'4iihmt\mW4k  ^'''^^  ^^^  ^^   Iiimtonxloq 

renders,  at  the  same  time,  all  these  co-efTicients  poliliW,  tMi^iffifh- 
l4feffowiJlo%i  og^nd'al^'Oie^'^i^ei^teiii  tM^^te'^gre^efe^^pcSitiW^Toot 
X<y^i  tfej^^qfiat^dri    [  e-ln-i«9i    »viji;^ofi    6Yi^    ^oo'igob    biiiii     t^iiJ    lo 

oa  EfiT,f:lIf;86keoc<DtBficieiitseofi:^qi^tidli   ^IJi^^^'^fetll  tlpQ^j;fe\%T  no 
.pd«i|)ba'j:Kfili^.rbf~«^'tfito-^  tliea?fefo^e';rb^iIitheo'ireJiP  ^^ 

ii©f;,^  ^ia>$t8.'l)jejL\DegMii(3^  ^-B(i!ife-^i(Qm-ti^^t>^^quat'iomixs  orlJ  o)  jasl 

-sqija. /i    ei,  T  toft   osa    oW  ,  .axodrnuii   o'lijns   iii   ;Jiml[   (taiV^I^ail.t 
and  m  order  tnat  every  value  of  w,  correspondmg  to  each  of  the 

m  J'unll  Jf^ml    04h,'A    V    .ooxiod    lion    yi.^O.  Jf^ii)    diib  ,JmvI,  loi'i 
vames  oi   x   and  ar,   may^  be   negative,  it   is   necessary   that  the 

greatest  positive  value  of  x  should  be  less  than  the  value  ^otx\ 

Hence,    this  valii^^^^ii^*  i^f^a^^iu^^^ifcy'^iiii*  cr^^i^aq'^sfiive 

roots.      If  w^no\€§ubstifeu^|e^ki-^Hec^8ieH^f(fi--^%i  X  the  values 

x'  —  I,    x'  —  2,    x'  —S,  &<j^  ,oW!i^tijlr  ^ifVja^^i  i^l.ftpi^[.^i,4i  ^\]\ 

niake  X^.negatiye.^tlien  the  last  number  which  i;^iidered..,it  nosi- 
afoo'i   ovi-:ti8oq  oaP  lo  jirnif   loiToqua    oilJ    go    oj  T  bim  ov/    .-C 
tive   will    be    the    least   superior    limit  of  the   positive  roots  „  in 

^  ^flOlf^JjpO     9X1 J     iO 

^''"'^  numbers^  ^  ^^  _  ,^^^  _^  ^^^^  __  ,,^^  _  ,^ 

-iiommoord    griibml   iil    dqooz^x^i^E.moblsa    ai   boxliom   sIiIT 
Let  x^  -  5x^  -  Qx^  -  19^  +  7  =  0.       ''^^^'^^  "^^^'^^^^ 

i^o4s'^^  Ifei^et^r^ipii^^Je^^^^o^^^a^y^A^v  t^^^-^^coi^^.qg^^e 

of  writing   the   pri^p^(^^rQt2^l3L^..,^  ^^|;|pr^,^^  in  the  formation  of 
the   deri  ved  polynomials ;  and  we  have, 

-fli  bnB   loliaqua  mlj^  bng  ^jfmi[   -lohf^ai    odl  bnfi    ot  enlufnoi    -;^i 
.bsiobianoo  ^in^ohoimm  ,a)ooi.  eviiij^on   ^rli  'io   aiimil   loI'T^l 

— .  =  Qx^  —  15a:   —  6,noijiiupo  Yjm  ni  ^I  jVaV*^ 

^      1 

j^  ^ —  —  3;     oAmn  ov/     .0  =  Yx 

^•^  .0  '^  I    rioiiiujpo  Y/on   f,   ovxjd   Iluda    ©w 

The  question  is  bow  reduced,  to  finding,  the  smallest  ^ntir« 
number  which,  substituted  m  place  of  ar,  will  render  all  of 
these   polynomials  positive. 


858  ELEMENTS  OF  ALGEBRA.  ICHAP.  XI. 

It  is  plafn  that  2  and  every  number  >  2,  will  render  the 
polynomial   of  the  first   degree  positive. 

But  2,  substituted  in  the  polynomial  of  the  second  degree, 
gives  a  negative  result ;  and  3,  or  any  number  >  3,  gives  a 
positive  result. 

Now,  3  and  4,  substituted  in  succession  in  the  polynomial 
of  the  third  degree,  give  negative  results ;  but  5,  and  any 
greater  number,  gives  a  positive  result. 

Lastly.  5  substituted  in  X,  gives  a  negative  result,  and  so 
does  6  ;  for  the  first  three  terms,  x^  —  bx^  — -  (jx^,  are  equiva- 
lent to  the  expression  x^  {x  —  b)  —  Gx^,  which  reduces  to  0  when 
X  =  6]  but  X  =  7  evidently  gives  a  positive  result.  Hence  7,  is 
the  least  limit  in  entire  numbers.  We  see  that  7  is  a  supe- 
rior limit,  and  that  6  is  not ;  hence,  7  is  the  least  limit,  as 
above   shown. 

2.  Applying   this   method   to   the   equation, 

x^-^Sx^-  Sx^  -  2bx^  +  4a;  -  39  =  0, 
the   superior  limit  is  found   to   be   6. 

3.  We  find  7  to  be  the  superior  limit  of  the  positive  roots 
of  the   equation, 

a;5  _  5i4  _  13^3  4.  17^2  _  69  =  0. 

This  method  is  seldom  used,  except  in  finding  incommeu- 
surable  roots. 

Superior   Limit   of  Negative   Roots. — Inferior   Limit  of  Posi 
tive   and  Negative  Hoots, 

286.  Having  found  the  superior  limit  of  the  positive  roots, 
it  remains  to  find  the  inferior  limit,  and  the  superior  and  in- 
ferior  limits   of  the   negative   roots,  numerically  considered. 

First,  If,  in  any  equation, 

X  =  0,     we  make     x  =  — , 

y 

we   shall  have  a  new  equation    Y=0. 

Since  we  know,  from  the  relation    a?  =  — ,     that   the  greatest 

y 


CHAP.   XI.  CONSEQUENCES   OF   PRINCIPLES.  859 

positive  va'-ue  of  y  in  the  new  equation  corresponds  to  the  least 
positive  value  of  x  in  the  given  equation,  it 'follows,  tliat 

If  we  determine  the  superior  limit  of  the  positive  roots  of  the 
equation  Y  =r  0,  its  reciprocal  will  be  the  inferior  limit  of  the 
positive   roots   of  the  given   equation. 

Hence,  if  we  designate  the  superior  limit  of  the  positive 
roots   of  the   Equation   F"=  0  by  L\  we   shall   have  for   the  in- 

ferior   limit   of  the   positive   roots   of  the   given   equation,     — . 

Second^  If  in   the   equation 

X  =  0,  we  make  x  z=l  —  y^ 
which  gives  the  transformed  equation  ]P  =  0,  it  is  clear  that 
the  positive  roots  of  this  new  equation,  taken  with  the  sign 
— ,  will  give  the  negative  roots  of  the  given '  equation ;  there- 
fore, determining  by  known  methods,  the  superior  limit  of  the 
positive  roots  of  the  new  equation  Z^  =  0,  and  designating  this 
limit  by  L^\  we  shall  have  —  L^^  for  the  superior  limit,  (nu- 
merically), of  the  negative  roots  of  the  given  equation. 

Third,  If  in  the  equation 

X  =  0,     we  make     x  = , 

we  shall  have  the  derived  equation  F'^  =  0.  The  greatest  posi- 
tive value  of  y  in  this  equation  jvill  correspond  to  the  least 
negative  value  (numerically)  of  x  in  the  given  equation.  If, 
then,  we  find  the  superior  limit  of  the  positive  roots  of  the 
equation   Y^^  =  0,    and   designate   it   by  Z^^^,  we    shall   have   the 

inferior  limit  of  the  negative  roots  (numerically)  equal  to  —  yyj^ 

Consequences  deduced  from  the  preceding  Princip^.es, 

First, 

287.  Every  equation  in  which  there  are  no  variations  in  the  signs, 
that  is,  in  which  all  the  terms  are  positive,  must  have  all  of  its  real 
roots  negative;  for,  every  positive  number  substituted  for  x,  will 
render  the  first  member  essentially  positive. 


860  ELEMENTS  OF  ALGEBRA.  [CHAP.  XL 

Second, 

288 •  Every  ( )mplete  equation^  having  its  terms  alternately  po.^l 
li  ve  and  negative,  must  have  its  real  roots  all  positive  ;  for,  every 
negative  number  substituted  for  x  m  the  proposed  equation,  would 
render  all  the  terms  positivo,  if  the  equation  be  of  an  even  de 
gree,  and  all  of  them  negative,  if  it  be  of  an  odd  degree.  Hence, 
their  sum  could  not  be  equal  to  zero  in  either  case. 

This  principle  is  also  true  for  every  incomplete  equation,  in  which 
there  results,  by  substituting  —  y  for  x,  an  equation  having  all  its 
terms  affected  with   the  same  sign. 

Third. 

289.  Every  equation  of  an  odd  degree,  the  co-efficients  of  which 
are  real,  has  at  least  one  real  root  affected  with  a  sign  contrary  to 
that  of  its  last  term. 

For,  let 

^m  ^  p^m-l  +   ,  ,  ,    Tx  zt  17=  0, 

be  the  proposed  equation  ;  and  first  consider  the  case  in  which 
the  last  term  is  negative. 

By  making  rr  =zz  0,  the  first  member  becomes  —  U.  But  by 
giving  a  value  to  x  equal  to  the  greatest  co-efiicient  plus  1,  or 
{K-\-  1),  the  first  term  x^  will  become  greater  than  the  arith- 
metical sum  of  all  the  others  (Art.  282),  the  result  of  this  sub- 
stitution will  therefore  be  positive;  hence,  there  is  at  least  one 
real  root  comprehended  between  0  and  ^-f-  1,  which  root  is  posi- 
tive, and  consequently  affected  with  a  sign  contrary  to  that  of  tlia 
last  term  (277). 

Suppose  now,  that  the  last  term  is  j^ositive 

Making  a;  ==  0  in  the  first  member,  we  obtain  -f  U  foi  the  rcsuit ; 
but  by  putting  —  {IC  ~\-  1)  in  place  of  .t,  we  shall  obtain  a  negor 
tive  result,  since  the  first  term  becomes  negative  by  this  sab 
stitution  ;  hence,  the  equation  has  at  least  one  real  root  com 
prehended  between  0  and  -—  (iiT-f  1),  which  is  negative,  oj 
(fffected  with  a  sign  contrary  to  that  of  the  last  U?rm. 


CHAP.   XI.  1  CONSEQUENCES   OF   PRINCIPLES.  361 

Fourth, 

290.  Every  equation  of  an  even  degree^  which  involves  only  rea. 
co-efficients,  and  of  which  the  last  term  is  negative,  has  at  least  two 
real  roots,  one  positive  and  the  other  negative. 

For,  let  —  U  he  the  last  term ;  making  x  =  0,  there  results 
—  U.  Now,  substitute  either  K+  I,  or  —  (if  +  1),  K  being 
the  greatest  co-efficient  in  the  equation.  As  m  is  an  even  number, 
the  first  term  x^  will  remain  positive ;  besides,  bj  these  substi- 
tutions, it  becomes  greater  than  the  sum  of  all  the  others ;  there- 
fore,  the  results  obtained  by  these  substitutions  are  both  positive, 
or  affected  with  a  sign  contrary  to  that  given  by  the  hypothesis 
X  =  0  ;  hence,  the  equation  has  at  least  two  real  roots,  one  positive, 
and  comprehended  between  0  and  1^+  I,  the  other  negative,  and 
comprehended  between  0  and  —  {K  -\-  1)  (277). 

Fifth. 

291.  If  an  equation,  involving  only  real  co-eflcients,  contains  imagi- 
nary roots,  the  number  of  such  roots  must  he  even, 

Eor,  conceive  that  the  first  member  has  been  divided  by  all  the 
simple  factors  corresponding  to  the  real  roots;  the  co-efficients 
of  the  quotient  will  be  real  (Art.  246);  and  the  quotient  must  alsc 
he  of  an  even  degree ;  for,  if  it  was  uneven,  by  placing  it  equal 
to  zero,  we  should  obtain  an  equation  that  would  contain  at  least 
one  real  root  (289) ;  hence,  the  imaginary  roots  must  enter 
by  pairs. 

Remark. — There  is  a  property  of  the  above  polynomial  quotient 
which  belongs  exclusively  to  equations  containing  only  imaginary 
roots ;  viz.,  every  such  equation  always  remains  positive  for  any 
real  value  substituted  for  x. 

For,  by  substituting  for  x,  K  -\- \,  the  greatest  co-efficient 
plus  1,  we  could  always  obtain  a  positive  result;  hence,  if  the 
polynomial  could  become  negative,  it  would  follow  that  when 
placed  eaual  to  zero,  there  -R^ould  be  at  least  one  real  roo\  com- 


362  ELEMENTS   OF   ALGEBRA.  [CHAP.   XI. 

prehended  between  X+  1  and  the  number  which  would  give  a 
negative  result  (Art.  277). 

It  also  follows,  that  the  last  term  of  this  polynomial  must  be 
positive,  otherwise  x  =  0  would  give  a  negative  result. 

Sixth, 

29.2 •  W?ien  the  last  term  of  an  equation  is  positive,  the  number 
of  its  real  positive  roots  is  even ;  and  when  it  is  negative,  the 
number  of  such   roots   is  uneven, 

For,  first  suppose  that  the  last  term  is  -f-  C/",  or  positive.  Since 
by  making  a;  ==  0,  there  will  result  +  U,  and  by  making  x  —  K  +  l, 
the  result  will  also  be  positive,  it  follows  that  0  and  K  -\-\ 
give  two  results  affected  with  the  same  sign,  and  consequently 
(Art.  279),  the  number  of  real  roots,  if  any,  comprehended  be- 
tween them,  is  even. 

When  the  last  term  is  —  U,  then  0  and  K  -\-  \  give  two 
results  affected  with  contrary  signs,  and  consequently,  they  com- 
prehend either  a  single  root,  or  an  odd  number  of  them. 

The  converse  of  this  proposition  is  evidently  true. 

Descartes'  Rule, 

293,  An  equation  of  any  degree  whatever,  cannot  have  a  greater 
number  of  positive  roots  than  there  are  variations  in  the  signs  of 
its  terms,  nor  a  greater  number  of  negative  roots  than  there  are 
permanences  of  these  signs, 

A  variation  is  a  change  of  sign  in  passing  along  the  terms.     A 

permanence  is  when  two  consecutive  terms  have   the   same  sign. 

In   the    equation 

X  —  a  =  0, 

there   is   one  variation,  and  one   positive  root,   x  =z  a. 

And  in  the  equation  x  +  b  z=0,  there  is  one  permanence,  and 

one  negative  root,  x  =  —  b. 

If  these  equations  be  multiplied  together,  member  by  member, 

there  will  resuU  an  equation  of  the  second  degree, 
x^  —  a 


+  b 


x-ab)    _  ^^ 


CHAF.   XI.j  DESCARTES'   RULE.  863 

If  a  is  less  vhaii  b,  the  equation  will  be  of  the  first  form 
(Art.  117);  and  if  a  >  ^,  the  equation  will  be  of  the  dP-cond 
form ;    that  is, 

a  <Cb     gives     x^  -}-  2px  —  ^  =  0, 
aiid  a  >  6        "        x^  -'  2px  —  q  =z  0, 

In  the  first  case,  there  is  one  permanence  and  one  yariation, 
and  in  the  second,  one  variation  and  one  permanence.  Since 
in  either  form,  one  root  is  positive  and  one  negative,  it  fol- 
lows that  there  are  as  many  positive  roots  as  there  are 
variations,  and  as  many  negative  roots  as  there  are  perma- 
nences. 

The  proposition  will  evidently  be  demonstrated  in  a  general 
manner,  if  it  be  shown  that  the  multiplication  of  the  first  mem- 
ber of  any  equation  by  a  factor  a:  —  a,  corresponding  to  a  posi- 
live  root,  introduces  at  least  one  variation,  and  that  the  multi- 
plication by  a  factor  x  -{-  a,  corresponding  to  a  negative  root, 
introduces  at  least  one  permanence. 

Take  'the   equation, 

^m  ±  J^^m^l  _±-  ^^m-2  -j.   (T^m-S  d=    .    .    .     ±  TiT  ±   U  =  0, 

:n  which  the  signs  succeed  each  other  in   any  manner  whatever. 
By  multiplying  by  x  —  a,  we  have 

^zTa 

The  co-efficients  which  form  the  first  horizontal  line  of  this 
product,  are  those  of  the  given  equation,  taken  with  the  same 
signs ;  and  the  co-efficients  of  the  second  line  are  formed  from 
those  of  the  first,  by  multiplying  by  a,  changing  the  signs,  and 
advancing   each   one   place   to   the   right. 

Now,  so  long  as  each  co-efficient  in  the  upper  line  is  greater 
than  the  corresponding  one  in  the  lower,  it  will  determine  the 
sign  of  the  total  co-efficient ;  hence,  in  this  case  there  will  be, 
fi'om  the  first  term  to  that  preceding  the  last,  inclusively,  the 
same  variations  and  the  same  permanences  as  in  the  proposed 
equation ;  but  the  last  term  zp  Ua  having  a  sign  contrary  to  that 
which  immediately  precedes  it,  there  must  be  one  more  varia- 
tion   than   in   the  proposed   equation. 


—  a 


x'^doB 
:=pAa 


zfBa 


p  ) 


364  ELEMENTS   OF   ALGEBRA.  [CHA'^.'^Si. 

When  a  co-efficient  in  the  lower  line  is  affected  with'  a^^sign 
contrary  to  the  one  corresponding  to  it  in  the  upper,  and'^'is 
also  greater  than  this  last,  there  is  a  change  from  a  perni^^ 
nence  of  sign  to  a  variation ;  for  the  sign  of  the  term  in  whicn 
this  happens,  being  the  same  as  that  of  the  inferior  co-efficient, 
must  be  contrary  to  that  of  the  preceding  term,  which  has 
been  supposed  to  be  the  same  as  that  of  the  superior  co-effi- 
cient. Hence,  each  time  we  descend  from  the  upper  to  the 
lower  line,  in  order  to  determine  the  sign,  there  is  a  variation 
which  is  not  found  in  the  proposed  equation  ;  and  if,  after 
passing  into  the  lower  line,  we  continue  in  it  throughout,  we 
shall  find  for  the  remaining  terms  the  same  variations  and  the 
same  permanences  as  in  the  given  equation,  since  the  co-efficients 
of  this  line  are  all  affected  with  signs  contrary  to  those  of  the 
primitive  co-efficients.  This  supposition  would  therefore  give  us 
one  variation  for  each  positive  root.  But  if  we  ascend  from 
the  lower  to  the  upper  line,  there  may  be  either  a  variation 
or  a  permanence.  But  even  by  supposing  that  this  passage  pro- 
duces permanences  in  all  cases,  since  the  last  term  =f  Ua  forms 
a  part  of  the  lower  line,  it  will  be  necessary  to  go  once  more 
from  the  upper  line  to  the  lower,  than  from  the  lower  to  the 
jpper.  Hence,  the  new  equation  must  have  at  least  one  more 
variation  than  the  pj^oposed ;  and  it  will  be  the  same  for  each 
positive   root  introduced   into   it. 

It  may  be  demonstrated,  in  an  analogous  manner,  that  the 
multiplication  of  the  first  inemher  hy  a  factor  x  -}-  a,  correspond- 
ing to  a  negative  root,  would  introduce  one  permanence  more. 
Hence,  in  any  equation,  the  number  of  positive  roots  cannot  be 
greater  than  the  number  of  variations  of  signs,  nor  the  number 
of  negative   roots  greater  than  the  number  of  permanences. 

Co7isequence.  ^ 

294.  When  the  roots  of  an  equation  are  all  real,  the  number 
of  podtive  roots  is  equal  to  the  number  of  variations,  and  the  num- 
ber of  negative  roots  to  the  number  of  permanences. 


CaXP/lXKPJ  DESCAKTES'   RULE.  865 

For,  J,9fe^???,;^en(^tei.,,tfie  degree  of  the  eqaation,  n  the  number 
of  variations  of  the  signs,  p  the  number  of  permanences ;  then, 

-liiijfi  oloiif/  orrj  Klnoloflh  = 

m  —  n  -\-  p. 

Moreover,  let  n'  denote  the  number  of  positive  roots,  and  p' 
the   number  of  negative   roots,  we  shall  have 

m  =  n'  -\-  p^ ; 
whejice,         w  +^  =  w'  4-^^,     or,     n  —  n'  z=  p^  —  p, 

^ow,  we  have  just  seen  that  n/  cannot  be  >  n^  nor  can  it  be 
les|;  M^^^  cM^^BP^>5^;-^46refore,  we   must  have 
Fioraoosd  Tiloifcf^^:)')  adti  ^   p^  =i  p, 

Eemark.— Wh^n  ano  ^uation  wants  some  of  its  ter;ns,  we  can 
often  discover  the  presence  o'f  ini aginary  roots,  by  means  of  the 
abo^eni^^ffXiij  hn&  ^^-'"^   \(\  fe-isdmon. 

For  exarnple,  take  the  equation 

x^  -\-  px  +  q  =^  0^ 
p  'kiidb^?(b§diEig^  0SBeiiijiaMyp^O8idiVeY>  intrfaducing   the  term  which 
is-tw#itk%;iby)Ra;ffeotingBTil  jxdth.  oihkucqpisSibiBnt  ±  0 ;    it  becomes 
•^*»  ^i3ii)o  dor>o  oi  dcfe^s^Or.  flf'ffvf- ^fi+iTg  i^fCbo 

t(gp  cfe4kM%  dfe5^^  t<?fe^§ft$Moi^%n,^%W%hould  obtain  only 
p^§a\n%kfe*',%TOr^as  cf^  inf^i^^  ^Jgfl^giV^^W^  variations.  This 
prO¥e^UHitr'tTiS'^S^41\ion^life^f^ii$4<>iiiYa  for,  if  they 

were  SiWkmm  MH,  #^#dTiM  B§']fftfe§s^?^^'»tue  of  the  supe- 
r[6¥^%^i9,'^Hmi'^t\i^  yt^W^S  8M^Ugmv4,^'am;>(hj^  virtue  of  the 
infeifr6^>'^§igtf,^  tfirft  Wb  ^^^&^^i^}M^<liP^s[mk  tkd  one  nega- 

tiv'(?,'^4fh{yh  ^^yntm^M^^mMm  ^^^  ^-^^^w   m 

-W^^ar>c6^li'd?^%(5*n^'>ft^ii  ai  ^^rfti^tf^-^f-^'tlft  form 

for,  mtroducmg  the  term    ±  0 .  x\  it  becomes 

noil^ijpo  JfiTpiie^g'^^^^  ojfiit  .OBjBp^od)  gnxod  shlT 

,  .  ,  ,0  r.:.T\  f  y:^  4-  %.^  j-  ^^^  4:^.  ^  .  ^"-*^^J.Qt>  ^-««v:a  ±  '    . 
which'  contains  one  permanetice  and   two   varmtions,  wficrher  %e 

taffe¥*^t3ierf^(i]^i^  ■\51v'4Mfe¥M  s!g^^""Th%1^fb^ey.%i^^eq^fetJ^i^^liiay 

have  its  three  roots  real,   v;z.,  two   positi^^   and   onfe  iie^i^fve  ; 

or,  two  of  its  Toom^m^m^Hm^nkiP^^^SiS&  Qii^n^M^,^^^^^ 

its(l^st.terma$r:p\fei4ve*X^^»^99).^»Si4-  .  .  .  4  ^-"'v/vf^tt 


366  ELEMENTS   OF  ALGEBRA.  LCHAP.    XL 

Of  the  commensurable  Boots  of  Numerical    Equattoi.s, 

295.  Every  equation  in  which  the  co-efficients  aie  whole  num- 
bers, that  of  the  first  term  being  1,  will  have  whole  numbersi 
only  for  its  commensurable  roots. 

For,  let  there  be  the  equation 

in  which  P,   Q  ,  ,  ,  T^   U^  are  whole  numbers,  and  suppose  that 

a 

It  were  possible  for  one  root  to  be   an   irreducible   fraction   -7-. 

Substituting  this  fraction  for  a:,  the  equation  becomes 

rim  fitn — 1  ftm — 2  /» 

whence,  multiplying  both  members  by  J/^^,  and  transposing, 

ft*'* 

---  =  —  Pa*^!  —  Qa'^'^b  —  ...   —  Tah^'^  —  Uh^^, 

0 

But  the  second  member  of  this  equation  is  composed  of 
the  sum  of  entire  numbers,  while  the  first  is  essentially  frac- 
tio;ial,  for  a  and  b  being  prime  with  respeet  to  each  other,  a'^ 
and  b  will  also  be  prime  with  respect  t^  each  other  (Art.  95), 
and  hence  this  equality  cannot  exist;  for,  an  irreducible  frac- 
tion cannot  be  equal  to  a  whole  number.  Therefore,  it  is  im- 
possible  for  any  irreducible  fraction  to  satisfy  the  equation. 

Now,  it  has  been  shown  (Art.  262),  that  an  equation  con- 
taining rational,  but  fractional  co-efficients,  can  be  transformed 
into  another  in  which  the  co-efficients  are  whole  numbers, 
that  of  the  first  term  being  1.  Hence,  the  search  for  commensU' 
table  roots,  either  entire  or  fractional,  can  always  be  reduced  to 
that  for  entire  roots, 

296.  This  being  the  case,  take  the  general  equation 

g,m  +  p^m-l  ^   ^^»_2    ^    ,    ^    J^  Jlx^^  S^J^  TX-^    IT  ==  0, 

and  let  a  denote  any  entire  number,  positive  or  negative,  whicli 
will  satisfy  it. 
Since  a  is  a    oot,  we  shall  have  the  equation 
am  4    Pa^^   i-  .  .  .  +  i^a3  -f-  5a2  +  ^a  -f  U^=  0  -     (!)• 


CHAP.   XI.]      COMMENSURABLE   ROOTS  OF  EQUATIOKS.  867 

Now  replace  a  by  all  the  entire  numbers,  positive  and  negative, 
between  1  and  the  limit  +A  and  between  —1  and  —U^\  those 
which  verify  the  above  equality  will  be  roots  of  the  equation. 
But  these  trials  being  long  and  troublesome,  we  will  deduce  from 
equation  (1),  other  conditions  equivalent  to  this,  and  more  easily 
applied. 

Transposing  in  equation  (1)  all  the  terms  except  the  last,  and 
dividing  by  a,  we  have, 

-  =  -  a^^i  -  Pa'«-2  _   .  .  .  _  ig«2  __  5«  _  /^  .     .     .     (2). 

Now,  the  second  member  of  this  equation  is  an  entire  number ; 

hence,  —  must  be  an  entire  number ;  therefore,  the  entire  roots  of 

the  equation  are  comprised  among  the  divisors  of  the  last   term. 
Transposing  —  ^  in  equation  (2),  dividing  by  a,  and  making 

—  +  r  =  r,     we  have, 

T 

—  =  -  a'^2  _  p^m-3  .  .  .  -  ^a  -  ASf     .     .     -     -     (3), 

T' 
The  second  member  of  this  equation  being  entire,    — ,  that  is, 

a 

the   quotient  of 

^+T    hy    a, 

IS  an  entire  numbers. 

Transposing  the  term  —  S  and  dividing  oy  a,  we  have,  by 
supposing 

±-  +  S=S% 
a 

—  =  -  a»«-3  ~  Fa'^  —  .  .  .  —  i?  . .  .  (4). 

The  second  member  of  this  equati:>n  being  entire,    — ,  that  i«, 

the  quotient  of 

T 

-  +  ^    by    a, 

ii  an  entire  number. 


368  ELEMENTS   OF   ALGEBRA.   •  [CHAP.   XI. 

By  continuing  to  transpose  the  terms  of  the  second  member 
into  the  first,  we  shall,  after  m  —  1  transformations,  obtain  an 
equation   of  the   form, 

a 
Tlien,   transposing   the  term   —  P,  dividing  by  a,  and  making 

^-\-P  =  P\     we  have     —=-1,     or h  1  ==  0. 

a  a  a 

This  equation,  which  results  from  the  continued  transforma- 
tions of  equation  (1),  expresses  the  last  condition  which  it  is 
requimte  for  the  entire  nilmber  a  to  fulfil,  in  order  that  it  ma^* 
be   known  to  be  a  root  of  th6  equation. 

297.  From  the  preceding  conditions  we  conclude  that,  when 
an  entire  number  a,  positive  or  negative,  is  a  root  of  the  given 
equation,  the  quotient  of  the  last  term^  divided  hy  a,  is  an 
entire  number. 

Adding  to  this  quotient  the  co-efHcient  of  a;\  the  sum  will 
he   exactly  divisible   hy  a. 

Adding  the  co-efficient  of  x^  to  this  last  quotient,  and  again 
dividing   by  a,  the  new  quotient  must  also  he  mitire ;  and  so  on. 

Finally,  adding  the  co-efficient  of  the  second  term,  that  is,  of 
a;^-\  to  the  preceding  quotient,  the  quotient  of  this  sum  divided 
hy  a,  must  he  equal  to  —  1  ;  hence,  tlte  result  of  the  addition  of 
1,  which  is  the  co-efficient  of  x™,  to  the  preceding  quotient,  must 
be  equal  to  0. 

Every  number  which  will  satisfy  these  conditions  will  be  a 
root,  and   those  which   do   not   satisfy  them  should  be  rejected. 

All  the  entire  roots  may  be  determined  at  the  same  time, 
by  tJie  following 

RULE. 

Aft^r  having  determined  all  the  divisors  of  the  last  term,  ivrite 
those  which  are  comprehended  between  the  limits  -{-  L  and  —  TY^ 
upon  the  same  horizontal  line  ;  then  underneath  these  divisors  write 
the   quotients   of  the    last   term   by   each   of  therii^^m\m  ytilw^  «»  u 


GHAP.   XI.] 


COMMENSURABLE  ROOTS. 


369 


Add  the  co-efficient  of  x^  to  each  of  these  quotients^  and  write 
the  sums  underneath  the  quotients  which  correspond  to  them. 
Then  divide  these  sums  hy  each  of  the  divisors^  and  write  the  quo* 
iients  underneath  the  corresponding  sums,  talcing  care  to  reject  the 
fractional  quotients    and    the   divisors   which  produce    them ;  and 


so   on. 


When  there  are  terms  wanting  in  the  proposed  equation, 
their  co-efficients,  which  are  to  be  regarded  as  equal  to  0,  must 
be  taken  into   consideration. 


EXAMPLES. 

I 
1.  What  are  the   entire   roots   of  the   equation, 
a;4  —  a;3  —  13^2  +  le^—  48  =  0  ? 

A  superior  limit  of  the  positive  roots  of  this  equation  (Art. 
284),  is  13  +  1  =  14.  The  co-efficient  48  need  not  be  con- 
sidered, since  the  last  two  terms  can  be  put  under  the  form 
16  (a;  —  3)  ;  hence,  when  a:  >  3,  this  part  is  essentially  positive. 

A   superior  limit  of  the  negative  roots  (Art.  286),  is 

-(1+/48),     or     -8. 

Therefore^  the  divisors  of  the  last  term  which  may  be  roots, 
are  1,  2,  3,  4,  6,  8,  12 ;  moreover,  neither  +  1,  nor  —  1,  will 
satisfy  the  equation,  because  the  co-efficient  —48  is  itself  greater 
than  the  sum  of  all  the  others :  we  should  therefore  try  only 
the  positive  divisors  from  2  to  12,  and  the  negative  divisors  from 
—  2  to  —  6  inclusively. 

By  observing  the  rule  given  above,  we  have 


12,         8,        6,  4,         3, 

_    4^   -    6,  --  8,  -  12,   -  16 
+  12,  +  10,  +  8,  +    4, 


f    1, 
-12, 

~    1, 
-    2, 


+    1, 
-12, 

-  3, 

-  4, 

-  1, 


0; 

-'3 


2. 
-24 

-  ^ 

—  4, 
-17 


-2,-3,-4,-6 

+  24,  -f  16,  +  12,  +    8 
+  40,  -f  32,  +  28,  +  24 


-20, 

..,  -    7, 

-    4 

-33, 

..,-20, 
..,  +    5, 
■„+   4, 
..,    -    1, 

-17 

24 


870  ELEMENTS  OF  ALGEBRA.  [CHAP.  XI, 

The  first  line  contains  the  divisors,  the  second  contains  the 
quotients  arising  from  the  division  of  the  last  term  —  48,  by- 
each  of  the  divisors.  The  third  line  contains  these  quotients,  each 
augmented  by  the  co-efficient  +  16  ;  and  the  fourth^  the  quotients 
of  these  sums  bj  each  of  the  divisors;  this  second  condition 
excludes  the  divisors   -f  8,    +6,   and    ~  3. 

The  fifth  contains   the   preceding   line  of  quotients,  each    aug 
mented  by  the  co-efficient  —  13,  and  the  sixth  contains  the  quo  ' 
tients  of  these  sums  by  each  of  the  divisors ;  the  third  condition 
excludes  the  divisors  3,   2,    —  2,   and    —  6. 

Finally,  the  seventh  is  the  third  line  of  quotients,  each  aug 
mented  by  the  co-efficient  —  1,  and  the  eighth  contains  the  quo- 
tients of  these  sums  by  each  of  the  divisors.  The  divisors  +  4 
and  —  4  are  the  only  ones  which  give  —  1  ;  hence,  +  4  and 
—  4  are  the  only  entire  roots  of  the  equation. 

In   fact,  if  we   divide 

x^  —  x^  —  13.^2  +  le^  _  48^ 

by  the   product   {x  —  4)  (a;  -f-  4),   or   x^  — 16,    the   quotient   wiJ 
be  x^  —  X  ■\-  2t^   which   placed   equal  to  zero,  gives 

1  1      / TT 

^  =  -2-^-2/=^' 

therefore,   the   four  roots   are 

4,      -4,    i-  +  i./3Tr    and     1  -  l/^Tl. 

2.  What  are  the  entire  roots  of  the   equation 

a:4_5^3  +  25a;~21  =0? 
8.  What  are  the  entire  roots  of  the   equation 

\^;x^  -  19a;*  +  6a;3  +  \^x^  -  19a;  +  6  =  0? 

4.  What   iare   the   entire   roots  of  the   equation 

9a;«  +  30a;«  +  22a;*  +  10a;3  +  Ylx^  -  20a;  -f  4  =  0  ? 


CHAP,  xij  Sturm's  theorem.  871 

SturnCs  Theorem. 

298.  The  object  of  this  theorem  is  to  explain  a  method  of  de- 
ter'hiining  the  number  and  places  of  the  real  roots  of  equations 
involving   but   one   unknown   quantity. 

Let  X=0    .    w    .    .    (1), 

represent  an  equation  containing  the  single  unknown  quantity  x ; 
X  being  a  polynomial  of  the  m*^  degree  with  respect  to  or,  the 
co-efficients  of  which  are  all  real.  If  this  equation  should  have 
equal  roots,  they  may  be  found  and  divided  out  as  in  Art.  2G9, 
and  the  reasoning  be  applied  to  the  equation  which  would  result. 
We  will   therefore   suppose  JT  =  0   to  have  no   equal  roots. 

299.  Let  us  denote  the  first  derived  polynomial  of  X  by  X„ 
and  then  apply  to  X  and  X^  a  process  similar  to  that  for  find- 
ing their  greatest  common  divisor,  differing  only  in  this  respect, 
that  instead  of  using  the  successive  remainders  as  at  first  ob- 
tained,  we  change  their  signs,  and  take  care  also,  in  preparing  for 
the  division,  neither  to  introduce  nor  reject  any  factor  except  a 
positive   one. 

If  we  denote  the  several  remainders,  in  order,  afler  their  sisns 
have  been  changed,  by  Xj,  X3  .  .  .  X„  which  are  read  X  second, 
X  third,  drc,  and  denote  the  corresponding  quotients  by  Q^,  Q^ 
•  •  Qt^u  ^^^  ^3,y  then  form  the  equations 

X=X,Q,-^X,    ....    (2). 

Xi  =  X^Qc^  —  X^ 


-3r„-i  ^  Xf^Qn  —  Xn^i 


(3). 


Xf^-^  —  Xr—i^r^l  —  Xf^ 

Since  by  hypothesis,  X  =  0  has  no  equal  roots,  no  common 
divisor  can  exist  between  X  and  Xi  (Art.  267).  The  last  re- 
mainder —  X„  will  therefore  be  different  from  zero^  and  inde- 
'pendent  of  a?. 


372  ELEMENTS   OF  ALGEBRA.  [CHAP.   XL 

300.  Now,  let  us  suppose  that  a  number  p  has  been  substi 
tuted  for  x  in  each  of  the  expressions  X,  Xj,  Xj  .  .  .  X^i ; 
and  that  the  signs  of  the  results,  together  with  the  sign  of  X^, 
are  arranged  in  *  a  line  one  after  the  other :  also  that  another 
number  q,  greater  than  p,  has  been  substituted  for  x,  and  the 
signs  of  the  results  arranged  in   like   manner. 

27ien  will  the  number  of  variations  in  the  signs  of  the  first 
arrangement,  diminished  by  the  number  of  variations  in  those  of 
the  second,  denote  the  exact  number  of  real  roots  comprised  be- 
tween  p   and   q. 

301.  The  demonstration  of  this  truth  mainly  depends  upon 
the  three  following  properties  of  the  expressions  X,  X,  .  .  X„,  &c 

I.  If  any  number  be'  substituted  for  x  in  these  expressions,  it  is 
impossible  that  any  two  consecutive  ones  can  become  zero  at  the 
same   time. 

For,  let  X^i,  X«,  X„+i,  be  any  three  consecutive  expressions. 
Then   among  equations  (8),  we   shall  find 

from  which  it  appears  that,  if  X„_i  and  X„  should  both  become 
0  for  a  value  of  x,  X^+j  would  be  0  for  the  same  value ;  and 
since   the  equation  which  follows  (4)  must  be 

we  shall  have  X^+a  =  0  for  the  same  value,  and  so  on  until 
we  should  find  X,  =  0,  which  cannot  be ;  hence,  X^i  and  X, 
cannot  both  become  0  for  the   same  value  of  x. 

II.  By  an  examination  of  equation  (4),  we  see  that  if  X«  be- 
comes  0  for  a  value  of  x,  X,^^  and  X,+i  must  have  contrary 
signs  ,   that  is,  ^ 

J^  any  one  of  the  expressions  is  reduced  to  0  by  the  substi- 
tution of  a  value  for  x,  the  preceding  and  following  ones  will 
have  contrary  signs  for  the  same  value. 


CHAP,  xi.j       Sturm's  theorem.  873 

111.  Let  us  substitute  a  +  u  for  x  in  the  expressions  X  and 
Xi,  and  designate  by  U  and  Ui  what  they  respectively  become 
under  this  supposition.     Then  (Art.  264),  we  have 


w 


2 


U  =A  +A'u  +A''  —  +  &c. 
U,=A,  +  A\u  +  A","^  +  &c. 


-    -    (5), 


in  which  A^  A\  A^\  &c.,  are  the  results  obtained  by  the  sub 
stitution  of  c?.  for  ar,  in  X  and  its  derived  polynomials ;  a!id 
Ai^  A\^  &e.,  are  similar  results  derived  from  Xj.  If,  now,  a  be 
a  root  of  the  proposed  equation  X  =  0,  then  -4  =  0,  ar  d  since 
A'  and  A^  are  each  derived  from  Xj,  by  the  substitution  of 
a  for  ir,  we  have  A^  =  ^i,  and  equations  (5)  become 

U=A^u  +  A^--^^.         .    .    .    (6). 


U,=iA'    +  A\u      +  &c. , 

Now,  the  arbitrary  quantity  u  may  be  taken  so  small  that 
the  signs  of  the  values  of  U  and  JJ^  will  depend  upon  the 
signs  of  their  iirst  terms  (Art.  276) ;  that  is,  they  will  be  alike 
when  u  is  positive,  or  when  a  +  w  is  substituted  for  ar,  and  un- 
like when  u  is  negative  or  when  a  —  u  is  substituted  for  x. 
Hence, 

If  a  number    insensibly    less    than    one   of   the    real  roots    of 

"X.  =  0   be  substituted  for   x   in   X   and  Xi,  the  results  will  hav€ 

contrary  signs ;  and  if  a  number  insensibly  greater  than  this  root 
be  substituted^  the  results   will  have   the  so.me  sign, 

302t  Now,  let  any  number  as  ^,  algebraically  less,  that  is, 
nearer  equal  to  —  oo,  than  any  of  the  real  roots  of  the  seveia] 
equations 

X=0,     Xi  =  0  .  .  .  X^i  =  0, 

be  substituted  for  x  in  the  expressions  X,  Xi,  Xj,  &;c.,  and .  the 
signs  of  the  several  results  arranged  in  order ;  then,  let  x  be 
increased  by  insensible  degrees,  until  it  becomes  equal  to  h 
the  least  of  all   the   roots  of   the   equations.     As   there  is   no 


874  ELEMENTS   OF   ALGEBRA.  [CHAP.   XI. 

root  of  either  of  the  equations  between  k  and  A,  none  of  the 
signs  can  change  while  x  is  less  than  h  (Art.  277),  and  the 
number  of  variations  and  permanences  in  the  several  sets  of 
results,  will  remain  the  same  as  in  those  obtained  bj  the  first 
substitution. 

When  X  becomes  equal  to  A,  one  or  more  of  the  expressions 
X,  X,  &c.,  will  reduce  to  0.  Suppose  X„  becomes  0.  Then, 
as  by  the  first  and  second  properties  above  explained,  neither 
X„_i  nor  X„+i  can  become  0  at  the  same  time,  but  must  have 
contrary  signs,  it  follows  that  in  passing  from  one  to  the  other 
(omitting  X„  =:  0),  there  will  be  one  and  onli/  one  variation ; 
and  since  their  signs  have  not  changed,  one  must  be  the  same 
as,  and  the  other  contrary  to,  that  of  JTa,  both  before  and  after 
it  becomes  0 ;  hence,  in  passing  over  the  three,  either  just  before 
Xn  becomes  0  or  just  after,  there  is  one  and  onlj/  one  variation. 
Therefore,  the  reduction  of  X^  to  0  neither  increases  nor  di- 
minishes the  number  of  variations ;  and  this  will  evidently  be 
the  case,  although  several  of  the  expressions  JTi,  X^,  &c.,  should 
become   0   at   the   same   time. 

If  X  z=z  h  should  reduce  X  to  0,  then  h  is  the  least  real  root 
of  the  proposed  equation,  which  root'  we  denote  by  a ;  and 
since  by  the  third  property,  just  before  x  becomes  equal  to  a, 
the  signs  of  X  and  X^  are  contrary,  giving  a  variation,  and  just 
afler  passing  it  (before  x  becomes  equal  to  a  root  of  X^  z=z  0), 
the  signs  are  the  same,  giving  a  permanence  instead,  it  follows 
that   in   passing   this   root  a  variation  is  lost. 

In  the  same  way,  increasing  x  by  insensible  degrees  from 
X  =ia  -\-  u  until  we  reach  the  root  of  X  =  0  next  in  order,  it 
Is  plain  that  no  variation  will  be  lost  or  gained  in  passing  any 
of  the  roots  of  the  other  equations,  but  that  in  passing  this 
roc.t,  for  the  same  reason  as  before,  another  vaiiation  will  be 
lost,  and  so  on  for  each  real  root  between  k  and  the  number 
last  substituted,  as  g,  a  variation  will  be  lost  until  x  has  been 
increased  beyond  the  greatest  real  root,  when  no  more  can  be 
lost   €ir  gained.     Hence,   the   excess  of  the  number   of  variations 


CHAP.  XL]  Sturm's  theorem.  875 

obtained  by  the  substitution  of  h  over  those  obtained  by  the 
substitution  of  g^  will  be  equal  to  the  number  of  real  roots 
comprised  between   k  and  g. 

It  is  evident  that  the  same  course  of  reasoning  will  apply 
when  we  commence  with  any  number  p^  whether  less  than  all 
the  loots  or  not,  and  gradually  increase  x  until  it  equals  any 
other  number  q.  The  fact  enunciated  in  Art.  299  is  therefore 
established. 

303  •  In  seeking  the  number  of  roots  comprised  between  p  and  q, 
rfehould  either  p  or  q  reduce  any  of  the  expressions  Xj,  Xg,  &e., 
to  Oj  the   result    will   not   be   affected   by  their   omission,    since 
the  number  of  variations  will  be  the  same. 

Should  p  reduce  X  to  0,  then  p  is  a  root,  but  not  one  of  those 
sought ;  and  as  the  substitution  of  p  +  u  will  give  X  and  Xi 
the  same  sign,  the  number  of  variations  to  be  counted  will  not 
be   affected  by   the   omission  of  X  =z  0, 

Should  q  reduce  X  to  0,  then  q  is  also  a  root,  but  not  one 
of  those  sought ;  and  as  the  substitution  of  5'  —  u  will  give  X 
and  Xi  contrary  signs,  one  variation  must  be  counted  in  passing 
from   X  to   Xi. 

304*  If  in  the  application  of  the  preceding  principles,  we  ob- 
serve that  any  one  of  the  expressions  X„  Xj  .  .  .  &;c.,  X„  for 
instance,  will  preserve  the  same  sign  for  all  values  of  x  in 
passing  from  p  to  q,  inclusively,  it  will  be  unnecessary  to  use 
the  succeeding  expressions,  or  even  to  deduce  them.  For,  as 
X„  preserves  the  same  sign  during  the  successive  substitutions, 
it  is  plain  that  the  same  number  of  variations  will  be  lost 
among  the  expressions  X,  Xj,  &;c.  .  .  .  ending  with  X„  as  among 
all  including  X,.  Whenever  then,  in  the  course  of  the  division, 
it  is  found  that  by  placing  any  of  the  remainders  equal  to  0, 
an  equation  is  obtained  with  imfiginary  roots  only  (Art.  291), 
it  will  be  useless  to  obtain  any  of  the  succeeding  remainders. 
This  principle  will  be  found  very  useful  in  the  solation  of 
numerical   examples. 


376  ELEMENTS  OF  ALGEBRA.      *  [CHAP.  XI 

305.  As  all  the  real  roots  of  the  proposed  equation  are  neces- 
sarily included  between  —  oo  and  +  od,  we  may,  by  ascertain- 
ing  the  number  of  variations  lost  by  the  substitution  of  these, 
in  succession,  in  the  expressions  X,  X,  .  .  .  X^,  .  .  &c.,  readily 
determine  the  total  number  of  such  roots.  It  should  be  ob- 
served, that  it  will  be  only  necessary  to  make  these  substitu- 
tions in  the  first  terms  of  each  of  the  expressions,  as  in  this 
case  the  sign  of  the  term  will  determine  that  of  the  entire  ex- 
pression (Art.  282). 

Having  found  the  number  of  real  roots,  if  we  subtract  this 
number  from  the  highest  exponent  of  the  unknown  quantity,  the 
remainder  will  be  the  number  of  imaginary  roots  (Art.  248). 

306.  Having  thus  obtained  the  total  number  of  real  roots, 
we  may  ascertain  their  places  by  substituting  for  ar,  in  succes- 
sion, the  values  0,  1,  2,  3,  &;c.,  until  we  find  an  entire  num- 
ber which  gives  the  same  number  of  variations  as  -f  oo.  This 
will  be  the  smallest  superior  limit  of  the  positive  roots  in  entire 
numbers. 

Then  substitute  —  1,  —-2,  &c.,  until  a  negative  number  is 
obtained  which  gives  the  same  number  of  variations  as  —  oo. 
This  will  be,  numerically,  the  least  superior  limit  of  the 
negative  roots  in  entire  numbers.  Now,  by  commencing  with 
this  limit  and  observing  the  number  of  variations  lost  in  passing 
from  each  number  to  the  next  in  order,  we  shall  discover  how 
many  roots  are  included  between  each  two  of  the  consecutive 
numbers  used,  and  thus,  of  course,  know  the  entire  part  of  each 
root.  The  decimal  part  may  then  be  sought  by  Bom?  of  tlii* 
known  methods  of  approximation. 

EXAMPLES. 

1.     Let  &a?s  — 6a;-- 1  :^0  =sX. 

The  first  derived   polynomial   (Art.  264),  h 


CHAP.  XL]  Sturm's  theorem.  877 

and  since   we   may  omit  the   positive   factor  6,   w^ithout  affecting 
the  sign,  we  may  write 

Dividing  X  by  Xi,  we  obtain  for  the  first  remainder,  —4a:  —  1. 
Changing  its  sign,  we  have 

Multiplying  Xi  by  the  positive  number  4,  and  then  dividing 
by  Xi,  we  obtain  the  second  remainder  —  3 ;  and  by  changing 
its   sign 

+  3  =  X3. 

The   expressions  to  be  be  used  are  then 
X^Sx^-^Gx-l,     Xi  =  40:2-1,      X2  =  4a;+1,      X3  =  +  3. 

Substituting  —  od  and  then  +  oo,  we  obtain  the  two  following 
arrangements  of  signs : 

h h 3  variations, 

+  +  +  + 0        « 

there  are  then   three  real   roots. 

If,  now,  in  the  same  expressions  we  substitute  0  and  +  1, 
and  then  0  and  —  1,  for  or,  we  shall  obtain  the  three  following 
arrangements : 

For           ir=  +  l         H — I — 1-+         0  variations, 
«  a?=0 +  +         1  " 


a:=:-l  -  +  -+         3 


\ 


As  a;  =  +  1  gives  the  same  number  of  variations  as  -4-  00, 
and  a?  =  —  1  gives  the  same  as  —  00,  +  1  and  —  1  are  the 
araallest  limits  in  entire  numbers.  In  passing  from  —  1  to  0, 
two  variations  are  lost,  and  in  passing  from  0  to  +  \,  one 
variation  is  lost ;  hence,  there  are  two  negative  roots  betweeu 
—  1   and   0,  and  one  positive  root  between  0   and    +  1. 

2.  Let  2a;*  -  13a?2  +  \0x  -  19  =  0. 


S78  ELEMENTS  OF  ALGEBRA.  [CHAP.   XI 

If  we  deduce  X,  Xi,  and  X^,  we  have  the  three  expressions 
X  =    2x^-  ISx^  +  10a:  -  19, 
Xi=    4x^-lSx   +    5, 
X,=  13a:2- 15:»   4.38. 

If  we  place  X^  =  0,  we  shall  find  that  both  of  the  roots  of 
the  resulting  equation  are  imaginary ;  hence,  X^  will  be  positive 
for  all  values  of  x  (Art.  290).  It  is  then  useless  to  seek  for 
X3  and  X4. 

By  the  substitution  of  —  cx)  and  +  od  in  X,  Xi,  and  Xj,  we 
obtain  for  the  first,  two  variations,  and  for  the  second,  none; 
hence,  there  are  two  real  and  two  imaginary  roots  in  the 
proposed   equation. 

3.  Let  x^  -^x^  +  Sx  —  l  =  0. 

4.  ic*  —    a;3  —  3a:2  +  a;2  —  a;  —  8  =  0. 

5.  x^--2x^  +  1=0. 
Discuss   each  of  the  above   equations. 

307 •  In  the  preceding  discussions  we  have  supposed  the 
equations  to  be  given,  and  from  the  relations  existing  between 
the  co-efficients  of  the  different  powers  of  the  unknown  quan- 
tity, have  determined  the  number  and  places  of  the  real  roots; 
and,  consequently,  the  number  of  imaginary  roots. 

In  the  equation  of  the  second  degree,  we  pointed  out  the 
relations  which  exist  between  the  co-efficients  of  the  different 
powers  of  the  '  unknown  quantity  when  the  roots  are  real, 
and  when  they  are  imaginary  (Art.  116). 

Let  us  see  if  we  can  indicate  corresponding  relations  among 
the  CO  efficients  of  an  equation  of  the  third  degree. 

Let  us  take  the  equation, 

x^-{-Fx^-{- Qx+  U=0, 

and  by   causing  the  second    term    to  disappear   (Art.   263),   it 
will  take  the  form, 

x^  +px  +  5'  =  0. 


CHAP.  XV  cakdan's  rule.  879 

Hence^  we  have 

X  =:    x^  +px  +  q, 

Zj  =  —  2px  -  Sq, 

If.  order  that  all  the  roots  be  real,  the  substitution  of  cx)  for 
r  in  the  above  expressions  must  give  three  permanences;  and 
the  substitution  of  —  od  for  x  must  give  three  variations.  But 
the  first  supposition  can  only  give  three  permanences  when 

_  4p3  _  27g2  >  0  ; 

•hat  is,  a  positive  quantity,  a  condition  which  requires  that  p  ^^ 
negative. 

If,  then,  p  be  negative,  we  have,  for  a;  =  00, 

— 4j93  —  27 q^  >  0 ;     that  is,  positive  : 

or,  4p^  +  27^2  <-  0  ;     that  is,  negative : 

hence,  j"  "'"  or    *^  ^'     which  requires    that   p  be 

p3        qi 
negative,   and  that  jz^^\\   conditions  which  indicate  that  the 

roots   are  all  real. 

OardarHs  Rule  for  Solving  Cubic  Equations, 

308.  First,  free  the  equation  of  its  second  term^  and  we 
have  the  form, 

x^+px  +  q=:0    .....      (1). 
Take  x  =y  -\-  z\ 

then  a;3  =  y3  +  2:3  +  g^^  [y  +  z)\ 

or,  hj  transposing,  and   substituting  x  for  y  +  z,   we  have 
ic3_3y2.^_(y3  +  2;3)  =  0    -    -    -     (2); 
and   by   comparing   this  with   equation   (1),   we  have 
—  ^yz  =  p ;      and      y^  4  ^3  =  —  g. 


380  ELEMENTS  OF  ALGEBRA, 

From   the   1st,   we   have 


[CHAP.  XL 


pi 


which,   being  substituted  in  the  second,   gives 


27y= 


=  -q-, 


cr  cxcaring  of  fractions,   and  reducing 
Solving   this  trinomial  equation   (Art.  124),   we   have 

^=\/-i+\/t+S 

and   the  corresponding  value  of  z  is 

I  /     q  f¥~Z¥ 

'  =  V-2"VT  +  27- 

But  since  a?  =  y  +  2?,   we   have 

'=l/[-l-vW^)]-'v/[-f-vf^)^ 

This  is   called    Cardan's  formula. 

By  examining  the  above  formula,  it  will  be  seen,  tha.i  it  is 
>uapplicable  to   the  case,   when   the  quantity 

4  "^  27' 

under  the  radical  of  the  second  degree,  is  negative ;  and  hence 
is  applicable  only  to  the  case  where  two  of  the  roots  are  imag 
inary  (Art.  307). 

Having  found  the  real  root,  divide  both  members  of  the  givcL 
equation  by  the  unknown  quantity,  minus  this  root  (Art.  247); 
the  result  will  be  an  equation  of  the  second  degree,  the  roots 
o(  which  may  be  readily  found.  , 


CHAP,  xij  Horner's  method.  881 


EXAMPLES. 


1.  What  are  the   roots  of  the   equation 
a;3  ^Qx^  +  lOx=zS1 


Ans.  4,     1 +y^^      1~V^-1.      ' 

2.  What  are  the  roots  of  the   equation 

Ans.   3,     3+.y^=T,     3-y-l. 

3.  What  are   the  roots   of  the   equation 

a;3  _  7a;2  +  14a;  =  20  ] 

Ans.   5,     1  +/"=^,     1  ~V^-"3; 

Preliminaries   to  Horner^s  Method. 

309.  Before  applying  the  method  of  Horner  to  the  solution 
of  numerical   equations,  it  will   be   necessary  to    explain, 

1st.  A  modification  of  the  method  of  multiplication,  called 
the  method  by   Detached  Co-efficients : 

2d.  A  modification  of  the  method  of  division,  called,  also,  the 
method  by  Detached   Co-efficients : 

3d.  A  second  modification  of  the  method  of  division,  called 
Synthetical  Division :   and, 

4th.  The  application  of  these  methods  of  Division  in  the 
Transformation  of  Equations.  ' 

Multiplication   hy  Detached    Co-efficients. 

310.  When  the  multiplicand  and  multiplier  are  both  homo- 
geneous (Art.  26),  and  contain  but  two  letters,  if  each  be  ar- 
langed  according  to  the  same  letter,  the  literal  part,  in  the 
several  terms  of  the  product,  may  be  written  immediately,  since 
the  exponent  of  the  leading  letter  -^ill  go  on  decreasing  from 
lefl  to  right  by  a  constant  number,  and  the  sum  of  the  exponents 
'>f  both   letters  will   be  the   same,  in   each  of  the  terms. 


382  ELEMENTS  OF  ALGEBRA,  [CHAP.  XI. 

EXAMPLES. 

L  Let  it   be  required   to   multiply 

x^  +  x^7/  -f  xy^  -{-  y^    by    a?  —  y. 
Since  x^  x  x  =z  x^,   the   terms   of  the  product  will   be  of  the 
4th   degree,   and   since   the   exponents   of  x  decrease   by   1,  and 
those  of  y  increase  by  1,  we   may  write   the  literal  parts  thus, 
x^,     x^y,     x'^y^,     xy^,     y*. 
In  regard   to   the   co- efficients,  we   have, 
Co-efficients  of  multiplicand,     -     -    -     1  +  1  +  1-fl. 
"  "    multiplier,  -     1—1 

14-1  +  1  +  1 
-1-_1_1_1 


co-efficients  of  the   product,  -     -     1+0  +  0  +  0—1; 

and  writing  these  co-efficients  before   the   literal   parts   to    which 
they  belong,  we   have 

x^  +  0,xhj  +  0.  x^y^  +  0.xy^  —  y^  =  x^  —  y*. 

2.  Multiply         2a3  —  ^ab^  +  6P     by     2a-  -  5b\ 

In  this  example,  the  term  a?b  in  the  multiplicand,  and  ab  in 
the  multiplier,  are  both  wanting  ;  that  is,  their  co-efficients  are 
0.     Supplying  these    co-efficients,  and  we   have, 

Co-efficients  of  multiplicand,    -    2  +  0—    3+    5 
"  "    multiplier,    .    .    2  +  0—    5 

4  +  0-    6  +  10 

-10-    0+15-25 


co-efficients  of  the  product,  --    4  +  0  —  16+10  +  15—25. 
Hence,  the   product  is,     4a^  —  Ua^^  +  lOa^^  +  15a6*  —  255'. 
8.  Multiply     x^  —  Sx^  +  Sx  —  l     by     a;2  —  2a;  +  1. 

4.  Multiply     y^  —  ya  +  —  a^     by     y^  +  ya  -  -  -—  a^. 

Remark. — The  method  by  detached  co-efficients  is  also  appli- 
cable to  the  case,  in  which  the  multiplicand  and  multiplier  con- 
tain  but  a  single  letter.  The  terms  whose  co-efficients  are  zero 
must  be   supplied,  when   ^ranting,  as  in   the  previous  example?. 


CHAP.  XI.'  DETACHED   CO  EFFICIENTS.  883 

EXAMPLES. 

1.  What  is   the  product  of  a*  +  Sa^  +  I     by     a^  —  3 1 

2.  What   is   the   product  ofP  —  l     by     b  +  21 

Division   by  Detached  Co-efficients, 

311.  When  the  dividend  and  divisor  are  both  homogeneous 
and  contain  but  two  letters,  the  division  may  be  performed  by 
means  of  detached   co-efficients,  in   the  following   manner : 

1.  Arrange  the  terms  of  the  dividend  and  divisor  according 
to   a   common  letter. 

2.  Subtract  the  highest  exponent  of  the  leading  letter  of  the  divi- 
sor from  the  exponent  of  the  leading  letter  of  the  dividend,  and 
the  remainder  will  be  the  exponent  of  the  leading  letter  of 
the   quotient. 

3.  The  exponents  of  the  letters  in  the  other  terms  follow 
the  same  law  of  increase  or  decrease  as  the  exponents  in  the 
corresponding   terms  of  the   dividend. 

4.  Write  down  for  division  the  co-efficients  of  the  different 
terms  of  the  dividend  and  divisor,  with  their  respective  signs, 
supplying  the  deficiency  of  the  absent  terms  with  zeros. 

5.  Then  divide  the  co-efficients  of  the  dividend  by  those  of 
the  divisor,  after  the  manner  of  algebraic  division,  and'  prefix  the 
several   quotients    to    their   corresponding  literal  parts. 

EXAMPLES. 


2. 

Divide 

8a5- 

-  4:a^x  - 

-  2a3^2  4.  a2a;3   ^y 

4a2  -  x\ 

The  literal 

part 

will  be 

< 

a^ar,     ax^,     x^ ; 

and 

fcr  the 

numerical  co- 

•efficients. 

8-4- 

-2+1    4+0- 

1} 

8  +  0- 

-2           2-1 

-4 

+  1     . 

-4 

+  1 

884  ELEMENTS  OF  ALGEBRA.  [CHAF.  XI. 

hence,  the  true  quotient  is  2a^  —  d^x',  the  co-efficients  after  —l, 
being   each   equal   to   zero. 

3.  Divide    x^  —  Sax^  —  8a^x^  +  ISa^x  —  8a*   by   x^  —  2ax  — 2a». 

4.  Divide  10a*  —  27a^x  +  S4:a^x^  —  ISax^  —  8a;*  by  2a« 
—  Sax  4-  4:x\ 

Eemarz. — The  method  by  detached  co-efficients  is  also  appli- 
cable to  all  cases  in  which  the  dividend  and  divisor  contain  but 
a  single  letter.  The  terms  whose  co-efficients  are  zero,  must  be 
supplied,  when  wanting,  as  in   the  previous   examples. 

EXAMPLES. 

1.  Let  it  be  required  to   divide 

6a* —  96     by     3a  —  6.      ♦ 
The  dividend,  in  this  example,  may  be  written  under  the  form, 

6a*  -f  0  .  a3  +  0  .  a2  -f  0  .  a  —  96a0. 
Dividing   a*   by   a,    we    have   a^   for   the   literal   part   of   the 
first  term  of  the  quotient ;  hence,  the  form  of  the  quotient  is 
a^,     a^,     a,     a®. 
For   the  CO- efficients,  we  have, 

6  +  0  +  0  +  0- 96  II 3- 6 

6—12  2  +  4  +  8  +  16     quotient ; 

hence,  the  true  quotient  is, 

2a3  +  4a2  +  8a  +  16. 

Synthetical  Division. 

312.  In  the  common  method  of  division,  each  term  of  the 
divisor  is  multiplied  by  the  first  term  of  the  quotient,  and  the 
products  subtracted  from  the  dividend;  but  the  subtractions  are 
performed  by  first  changing  the  sign  of  each  product,  and  then 
adding.  If,  therefore,  the  signs  of  the  divisor  were  first  changed, 
we  should  obtain  the  same  result  by  adding  the  products,  instead 
of  subtracting  as  before,  and  the  same  for  any  subsequent  oper- 
ation. 


CHAP    XI.]  SYNTHETICAL  DlVISIOlir.  385 

By  this  process,  the  second  dividend  would  be  the  same  as 
by  the  common  method.  But  since  the  second  term  of  the  quo^ 
tient  is  found  by  dividing  the  first  term  of  the  second  dividend 
by  the  first  terni  of  the  divisor ;  and  since  the  sign  of  the  latter 
has  been  changed,  it  follows,  that  the  sign  of  the  second  term 
of  tbi  quotient  will  also  be  changed. 

To  avoid  this  change  of  sign,  the  sign  of  the  first  term  of  the 
divisor  is  left  unchanged,  and  the  products  of  all  the  terms  of 
the  quotient  by  the  first  term  of  the  divisor,  are  omitted;  be 
cause,  in  the  usual  method,  the  first  termj  in  each  successive 
dividend  are  cancelled  by  these  products. 

Having  made  the  first  term  of  the  divisor  1  before  commenc 
ing  the  operation,  and  omitting  these  several  products,  the  co-effi- 
cient of  the  first  term  of  any  dividend  will  be  the  co-efficient  of  the 
succeeding  term  of  the  quotient.  Hence,  the  co-efficients  in  the 
quotient  are,  respectively,  the  co-efficients  of  the  first  terms  of 
the  successive  dividends. 

The  operation,  thus  simplified,  may  be  fdrther  abridged  by 
omitting  the  successive  additions,  except  so  much  only  as  may 
be  necessary  to  show  the  first  term  of  each,  dividend ;  and  also, 
by  writing  the  products  of  the  several  terms  of  the  quotient  by 
the  modified  divisor,  diagonally,  instead  of  hoi  izon tally,  the  first 
product  falling  under  the  second  term  of  the  dividend. 

Hence,  the  following 

RULE. 

I.  Divide  the  divisor  and  dividend  hy  the  co-efficient  of  the  first 
term  of  the  divisor^  when  that  co-efficient  is  not  1. 

II.  Write^  in  a  horizontal  line,  the  co-efficients  of  the  dividend, 
with  their  proper  signs,  and  place  the  co- efficients  of  the  divisor, 
with  all  their  signs  changed,  except  the  first,  on  the  right. 

III.  Divide  as  in  the  method  hy  detached  co- efficients,  except  thai 
no  term  of  the  quotient  is  multiplied  hy  the  fi.rst  term  of  the  divi- 
sor, and  that  all  the  products  are  written  diagonally  to  the  right, 
under  the  terms  of  the   dividend  to  which  they  cof*respond^ 

25 


386  ELEMENTS  OF  ALGEBRA.  [CHAP.   XI. 

IV.  The  first  term  of  the  quotient  is  the  same  as  that  of  the 
dividend ;  the  second  term  is  the  sum  of  the  numbers  in  the  second 
column ;  the  third  term,  the  sum  of  the  numbers  in  third  column, 
aiid    so   on,  to  the  right, 

V.  When  the  division  can  be  exactly  madCy  columns  will  he  found 
at    the   right,  whose   sums  will   be  zero:   when  the  division   is  not 
exact,  continue  the  operation  until  a  sufficient   degree  of  approxi- 
mation is  attained.     Having  found  the  co-efficients,  annex  to  them  * 
the  literal  parts, 

EXAMPLES. 

1,     Divide 

a8  -  f>a^x  +  10a3a;2  —  lOaH^  +  bax^  —  x^    by     a^  —  2aa;  +  xK 

1-5  +  10-10  +  5-1  11  1  +  2-1 

2--    6+    6-2  1^3  +  3^1 

-    1+    3-3  +  1 


1  __  3  +    3  -    1      0      0. 

Hence,  the   quotient  is 

a?  —  Za'^x  +  3aa;2  —  x\ 

Remark. — The  first  term  of  the  divisor  being  always  1,  need 
not  be  written.  The  first  term  of  the  quotient  is  the  same  as 
that  of  the   dividend. 

2.     Divide 
aj6_5^5+i5^4_24a;3+27ir2-13ir+5     by     x^-2x^+^x'^-2x-\-\. 
1  _  5  +  15  _-  24  +  27  -  13  +  5  ||  1  +  2-4  +  2- 1 
+  2-    6  +  10  1-.3  +  5 

-    4  +  12-20 

+    2  -    6  +  10 

-1+3-5 


1--3+5        0        0        0      0. 

Hence,  the  quotient  is    a;^  —  3a:  +  5. 
3.     Divide 

a«  +  2a*6  +  ^W  .«  ^253  _  2a6*  -  35«    by    a»  +  2ab  +  35». 
Ans,   a3  +  0.a25  +  0.a62- 63_^3_  j3, 


CHAP    XI.  J  SYNTHETICAL   DIVISION.  887 

4.  Divide    I  —  x  hj  I  +  x.     Ans.  l  —  2x  +  2a;«  -  2x^+  &c. 

5.  Divide    1  hy  I  —  x.  Ans.  I  +  x  +  x'^  +  x^  +  &ic, 

0.  Divide    x'^  —]p  bj  x  —  y, 

Ans,  x^  +  x^y  +  x^y'^  +  x^y^  +  x'^y^  +  xy^  +  y®. 

7.  Divide    a^  —  3a*a;2  +  ^a^x^  —  x^  by  a^  —  Sa^  +  Saaj^  -  x\ 

Ans.  a3  +  Sa^a;  +  ^ax'^  +  a;3. 

313.  To  transform  an  equation  into  another  whose  roots  shall  be 
the  roots  of  the  proposed  equation,  increased  or  diminished  hy  a  given 
quantity. 

A  method  of  solving  this  problem  has  already  been  explained 
(Art.  264);  but  the  process  is  tedious.  We  shall  now  explain 
a  more  simple  method  of  finding  the  transformed  equation. 

Let  it  be  required  to  transform  the  equation 

ax"^  +  Px-^^  +  Qx'^'^  ....  Tx+  U=z  0 

into  another  whose  roots  shall  be  less  than  the  roots  of  this 
equation  by  r. 

If  we  write  y  +  r  for  x,  and  develop,  and  arrange  the  terms 
with  reference  to  y,  we   shall  have 

aytn  ^  pfytnr^l  +  qyfnr^'l   .    .    .    .    +  ^/y  +   JJ/  =:  0      -      .      -      (1). 

But  since  y  =.  x  —  r,   equation   (1),  may   take  the  form 

a{x-^rY+P\x^rY-^  +  Q\x  -  r)"»-2 'jyix  -r)+  U'=0    (2), 

which,  when  developed,  must  be  identical  with  the  given  equa- 
tion. For,  since  y  +  r  was  substituted  for  x  in  the  proposed 
equation,  and  then  ic  —  r  for  y  in  the  transformed  equation,  we 
must  necessarily  have  returned  to  the  given  equation.  Hence, 
we  have 

a{x  -  rY  +  P\x  ~  rY"^  +  Q'{x  -  r)"-*  .  .  .  T  (x -^r) +  17 
=  ax"^  +  Px'^^  +  Qx"^^  .  .  .  Tx+  Cr=  0. 

If  now  we  divide  the  first  member  by  x  —  r,  the  quotient 
will  be 

a{x  -  r)^^  +  P\x  -  r)^2  +  Qf^^  _  ;.)»i-3  .  .  ,  ^, 
and  the  remainder   U\ 


S88  ELEMENTS   OF   ALGEBEA.  ICHAP.   XI 

Bat  since  the  second  member  is  identical  v^  th  the  first,  the 
very  same  quotient  and  the  same  remainder  would  arise,  if  the 
second  member  were  divided  by  x  —  r\   hence, 

If  the  fir U  member  of  the  given  equation  he  divided  by  the  unknown 
quantity  minus  the  number  which  expresses  the  difference  between  th 
^oots,  the  remainder  will  be  the  absolute  term  of  the  transformed  equation. 

Again,  if  we  divide  the  quotient  thus  obtained :  viz., 
a{x  —r)'*-i  +  F'{x  —  y)"»-2  +  Q\x  —  r)^^  .  .  .  ^v 
by  X  —-  r,  the  remainder  will  be  T^,  the  co-efficient  of  the  term 
last  but  one  of  the  transformed  equation ;  and  a  similar  result 
would  be  obtained  by  again  dividing  the  resulting  quotient 
by  X  —  r.  Hence,  by  successive  divisions  of  the  poly- 
nomial in  the  first  member  of  the  given  equation  and  the  quo- 
tients which  result,  by  x  ■— r,  we  shall  obtain  all  the  co-efficients 
of  the  transformed  equation,    in  an    inverse  order. 

Remark. — When  there  is  an  absent  term  in  the  given  equation, 
rts  place  must  be  supplied  by  a  0. 

EXAMPLES. 

Transform  the   equation 

5x^  —  12rr3  +  Sx^  + 4x^5  =  0 
into  anot>*sr  whose  roots  shall  each  be  less  than  those  of  the  given 
^uatioD  %y  2. 

First  Operation, 
bx^  -  12a;3  +  3:g2  g.  4a;  ^  5  |[  x  -  2 
5ic*  — 10^3  5ar3  — 2a;2  — a;  +  2 


-    2x^  +  Sx^ 

^    2a;3  +  4:r2 

+  Ax 

-    x^ 

-    x^ 

+  2x 

2x^ 

-5 

2x- 

-4 

—  1     1st  remainder 


CHAP.   XI.  I  SYNTHETICAL  DIVISION.  889 

Second  Operation, 


5a;3_    2a;2-a?  +  2 
5a;3  -  \0x^ 


X  -2 


bx^  +  8a;  +  15 


8a;2- 

a? 

8a:2- 

-  16a? 

\bx  + 

2 

15a;-. 

30 

32    2d  remainder. 
Third  Operation,  Fourth  Operation, 


5x^  +    So;  +  15 
5ir2  —  10a; 


5a; +  18 


5a;  +18  5a;  -  10 


a;-2 


18a;  +  15  28    4th  remainder. 

18a;  -  36 


51     3d  remainder. 
Therefore,  the  transformed  equation  is 

5y*  +  28y3  +  51y2  +  32y  —  1  =  0. 
This  laborious   operation   can  be   avoided   by   the   synthetical 
method  of  division  (Art.  312). 

Taking  the  same  example,  and  recollecting  that  in  the  syn- 
thetical method,  the  first  term  of  the  divisor  not  being  used,  may 
be  omitted,  and  that  the  first  term  of  the  quotient,  by  which 
the  modified  divisor  is  to  be  multiplied  for  the  first  term  of  the 
product,  is  always  the  first  term  of  the  dividend ;  the  whole  of 
the  work  may  be  thus  arranged : 

5-12         +3         +4         -5[[2_^ 
10         -4         —2  4 


-    2 

-1 

2    -1 

10 

16 

30 

8 

15 

32   .'.  2^  =  32 

10 

36 

18 

51 

.-.  Q'  =  51 

10 

28 

.-.P' 

=  28; 

890  ELEMENTS   OF   ALGEBRA.  lCHAP.   XI. 

for  it  is  plain  that  the  first  remainder  will  fall  under  the  abso- 
lute term,  the  second  under  the  term  next  to  the  left,  and  so 
on.     Hence,  the  transformed  equation  is 

5y*  -f  28y3  +  51^2  +  32y  —  1  =  0. 
2.  rind  the   equation  whose  roots  are  less  by  1.7  than  thos« 
of  the  equation 

^3  __  2a;2  +  3a;  —  4  =  0. 

First,  find  an  equation  whose  roots  are  less  by  1. 
1-2+3         -4[[1^ 
1-1  2 


•-1  2        -2 

1  0 

0  2 

1 

T 
We  have  thus  found  the  co-eflicients  of  the  terms  of  an  equa- 
tion whose  roots  are  less  by  1  than  those  of  the  given  equation  ; 
the   equation  is 

a;3  +  a?2  +  2a;  -  2  =  0  ; 
and   now   by  finding  a  new   equation  whose   roots   are  less  than 
those  of  the  last  by  .7,  we  shall  have  the  required  equation :  thus, 

1  +  1  +2  -2||.7 

.7  1.19  2.233 


1.7  3.19  .233 

.7  1.68 


2.4  4.87 

.7 


3.1 

hence,  the  required  equation  is 

2/3  +  3.1y2  +  4.87y  +  .233  =  0. 

This  latter  operation  can  be  continued  from  the  former,  witli- 
oat  arranging  the  co-efficients  anew.  The  operations  have  been 
explained  separately,  merely  to  indicate  the  several  steps  in  the 


CHAP.   XI.]  SYNTHETICAL  DIVISION.  391 

transformation^  and  to  point  out  the  equations,  at  each  step 
resulting  from  the  successive  diminution  cf  the  roots.  Com- 
bining the  two  operations,  we  have  the  following  arrangement: 
I      _2       +3         -4(1.7;    or,       1-2  +3       -4(1.7 

1-12  1.7       -  .51        4.233 

-    .3  2^  .233 

1.7  2.38 


-1 

2 

-2 

1 

0 

2.233 

0 

2 

.233 

1 

1.19 

1.7 

3.19 

.7 

1.68 

1.4  4.87 

1.7 


2.4        4.87 

.7 
3T 
We   see,  by  comparison,  that  the  above  results  are  the  same 
as  those   obtained  by  the   preceding   operations. 

3.  Find   the   equation  whose   roots   shall   be   less   by  1  than 

the  roots  of 

«3  -  7«  +  7  =  0. 

4.  Find   the   equation   whose  roots  shall  be  less    by   3    than 
the  roots  of  the   equation 

a;4  _  3^3  __  15^2  +  49^  _  12  =  0. 

Arts,  y*  +  9y3  +  I2y^  —  14y  =  0. 

5.  Find  the   equation   whose  roots   shall   be   less  by  10   than 
the  roots   of  the   equation 

x^  +  2x^  +  3a;2  +  4a;  -  12340  =  0. 

Ans.  y^  +  42y3  +  663y2  +  4664y  =  0. 

6.  Find  the   equation  whose  roots  shall  be  less    by  2  tlidu 
the  roots   of  the   equation 

x^  +  2a;3  —  (Sz^  -  10a;  =  0. 
Am.  ys   ;.  iQy*  ^  42^.':  4.  86y2  +  70y  +  4  =  0. 


392  ELEMENTS  OF  ALGEBRA.  [CHAP.   XI 

Horner's  Method  of  approxvnatmg  to  the  Beat  Boots  of 
Numerical  Equations, 

314i  The  nethod  of  approximating  to  the  roots  of  a  nnmeri 
cal  equation  of  any  degree,  discovered  by  thp  English  math©' 
niatician  W.  G.  Horner,  Esq.,  of  Bath,  is  a  /rocess  of  very 
remarkable   simplicity  and   elegance. 

The  process  consists,  simply,  in  a  succession  of  transforma 
tions  of  one  equation  to  another,  each  transformed  equation,  as 
it  arises,  having  its  roots  equal  to  the  difference  between 
the  true  value  of  the  roots  of  the  given  equation,  and 
the  part  of  the  root  expressed  by  the  figures  already 
found.  Such  figures  of  the  root  are  called  the  initial  Jigures, 
Let 

V=ix'^  + Px'^-^-\- Qx^-'^  .  .  .  .  +  Tx-\-U=zO    -    -    -     (1) 

be  any  equation,  and  let  us  suppose  that  we  have  foun*'"  a 
part  of  ^one  of  the  roots,  which  we  will  denote  by  m,  and  de- 
note  the   remaining   part   of  the   root  by  r. 

Let  us  now  transform  the  given  equation  into  another,  wLose 
roots   shall   be   less   by  m,  and  we   have  (Art.  313), 

V  z=:r^  +  F'r^^  -h  §V^2  .  .  _  +  ^V  +  C/""  =  0  •      (2). 

Now,  when   r   is   a   very  small   fraction,  all  the   terms  of  tho 

'  second  member,  except  the  last  two,  may  be  neglected,  and  the 

first  figure,  in   the  value  of  r,  may  be  found  from  the  equation 

U^  IT 

Tr-\-  C/'  =  0  ;    giving  -  r  =  — ;    or  r  =:  -  -—  ;    hence, 

The  first  figure  of  r  is  the  first  figure  of  the  quotient  obtained  by 
lividing  the  absolute  term  of  the  transformed  equation  by  the  penulti- 
mate co-efficient. 

If,  now,  we  transform  equation  (2)  into  another,  whose  roots 
shall  be  less  than  those  of  the  previous  equation  by  the  first 
figure  of  r,  and  designate  the  remaining  part  by  «,  we  shall 
have, 

V'  -xz^^-it  P''s^-    ^  Q^'s"^-^  .  .  .  .  +  T^'s  +  W'  =zO, 


CHAP.  XI.  1       Horner's  method. 

the  roots  of  which  will  be  less  than  those  of  the  given  equa- 
tion by  m  +  the  first  figure  of  r.  The  first  fig^are  in  the  value 
of  %  is  found  from  the  equation, 

T^s-^W^(),     giving     B=^. 

We  may  thus  continue  the  transformations  at  pleasure,  and 
each  one  will  evolve  a  new  figure  of  the  root.  Hence,  to  find 
the  roots  of  numerical   equations. 

I.  Find  the  number  and  places  of  the  real  roots  by  Sturms* 
theorem,  and   set  the  negative  roots  aside. 

II.  Transform  the  given  equation  into  another  whose  roots  shall 
be  less  than  those  of  the  given  equation,  by  the  initial  figure  or 
figures  already  found:  then,  by  Sturms''  theorern,  find  the  places 
of  the  roots  of  this  new  equation,  and  the  first  figure  of  each  will 
be    the  first  decimal  place   in   each  of  the   required  roots. 

III.  Transform  the  equation  again  so  that  the  roots  shall  be  less 
than  those  of  the  given  equation,  and  divide  the  absolute  term  of 
the  transformed  equation  by  the  penultimate  co-efficient,  which  is 
called  the  trial  divisor,  and  the  first  figure  of  the  quotient  will  be 
the  next  figure  of  the  root. 

IV.  Transform  the  last  equation  into  another  whose  roots  shall 
he  less  than  those  of  the  previous  equation  by  the  figure  last  found, 
and  proceed  in  a  similar  manner  until  the  root  be  found  to  the 
required  degree  of  accuracy. 

Remark  I. — This  method  is  one  of  approximation,  and  it  may 
happen  that  the  rejection  of  the  terms  preceding  the  penultimate 
term  will  affect  the  quotient  figure  of  the  root.  To  avoid  this 
source  of  error,  find  the  first  decimal  places  of  the  root,  also, 
by  the  theorem  of  Sturm,  as  in  example  4,  page  399,  and  when 
the  results  coincide  for  two  consecutive  places  of  decimals,  those 
Bubsequently  obtained  by  the  divisors  may  be  relied  on. 

Eemark  II. — When  jhe  decimal  portion  of  a  negative  root  is 
to  be  found,  first  transform  the  given  equation  into  ar.other  by 
changing  the  signs  of  the  alternate  terms  (Art.  280),  and  then 
find  the  decimal  part  of  the  corresponding  positive  root  of 
this  new   equation. 


894 


ELEMENTS  OF  ALGEBRA. 


[CHAP.   XI. 


IIL  When  several  decimal  places  are  found  in  the  root,  the 
operation  may  be  shortened  according  to  the  method  of  con^ 
tractions  indicated  in   the   examples. 

314#  Let  us  now  work  one  example  in  full.  Let  us  take  the 
equation   of  the   third   degree, 

x^-lx  +  lf  z=zO. 
By   Sturm's  rule,  we  have  the  functions  (Art.  299), 

X  =     a;3  -  7iC  +  7 

jri  =  3a;2-7  • 

X^  =  2x  --S 
X,=  +  1. 
Hence,  for  a;  =  oo,  we  have     +     +    +     +     no  variation, 

ir=— Qo"  —     +     —     +     three  variations ; 

tlierefore,  the  equation  has  three  real  roots,  two  positive  and  one 
negative. 

To  determine  the  initial  figures  of  these  roots,  we  have 

fora:  =  O...H h  for  a:  =      0...H h 

x=l...  + +  a;=-l...  + -h      • 

a?  =  2...+  +  +  +  a:=— 2...  +  +  — 4- 

ar=~3...+  +  -  + 
a:=-4...  -+  -  + 

hence   there  are   two  roots  between  1  and  2,  and  one  between 
—  3   and    —  4. 

In  order  to  ascertain  the  first  figures 
in  the  decimal  parts  of  the  two  roots 
situated  between  1  and  2,  we  shall  trans- 
form the  preceding  functions  into  others, 
in  which  the  value  of  x  is  diminished  by  2  —  4 

1.       Thus,    for    the   function  X,  we  have  1 

this  operation :  T 


1  +  0  -  7  +  7  (1 
1  +  1-6 

1  -6  +  1 
1+2 


And  transforming  the  others  in 
*he  same  way,  we  obtain  the 
tuncticns 


r,  =  3y«  +  6y  -4; 
r,  =  2y  -1; 
r,  =.  +  1. 


CHAP.   XI.] 

Horner's  method. 

895 

Let  y  = 

.1 

we  have     H f- 

two 

variations, 

y  = 

.2 

+  --  + 

(C 

y  = 

.3 

+ + 

(C 

y  = 

.4 

+ 

one 

variation, 

y  = 

.5 

-  -  =F  + 

(£ 

y  = 

.6 

-  +  +  + 

((                  « 

y  = 

.7 

+  +  -f  + 

no  ^ 

variation. 

Therefore  the  initial  figures  of  the  two  positive  roots  are  1.3,  1.6. 

\  et  us   now  find   the   decimal   part  of  the 

J   first   root. 

1       hO 

-7 

+  7  (1.356895867 

1 

1 

-6 

1 

-6 

*1 

1 

2 

-.903 

2 

*-.4 

**.097 

1 

.99 

-  .086625 

*3.3 

-3.01 

*«*  .010375 

8 

1.08 

% 

-  .009048984 

3.6 

**-1.93 

***  .001326016 

3 

.1975 

) 

-.001184430 

**3.9  5 

-1.7325 

.000141586 

5 

.2000 

) 

-  .000132923 

4.00 

***-!. 5325 

.000008663 

5 

.02433 

6 
4 

-  .000007382 

***4.0  56 

-1.50816 

.000001281 

6 

.024372 

-.000001181 

4.0  62 

****_  1.48379 

2 

.000000100 

6 

.00325 

4 

8  . 

-  .000000089 

•***4.0i68  8 

-1.48053 

.000000011 

8 

.00325 

4 

-.000000010 

|4.0l69e 

-1.4772 

8 

1 

.0003 

6 

-1.4769 

2 

0003 

6 

* 

-11.4|41716 

5 

896  ELEMENTS   OF  ALGEBRA.  [CHAP.    XI 

The   operations   in   the   examj)le   are  performed  as  follows : 

1st.  We  find  the  places  and  the  initial  figures  of  the  posi- 
tive  roots,  to  include  the  first  decimal  place  by  Sturms'  theorem. 

2d.  Then  to  find  the  decimal  part  of  the  first  positive  root, 
we  ^rrange  the  co-efficients,  and  perform  a  succession  of  trans 
formations  by  Synthetical  Division,  which  must  begin  with  the 
initial  figures  already  known. 

We  first  transform  the  given  equation  into  another  whose 
roots  shall  be  less  by  1.  The  co-efficients  of  this  new  equation 
are,  1,  3,  —4  and  1,  and  are  all,  except  the  first,  marked  by 
a  star.  The  root  of  this  transformed  equation,  •  corresponding 
to  the  root  sought  of  the  given  equation,  is  a  decimal  frac- 
tion  of  which  we   know   the   first   figure   3. 

We  next  transform  the  last  equation  into  another  whose 
roots  are  less  by  three-tenths,  and  the  co-efficients  of  the  new 
equation   are   each   marked   by   two   stars. 

The  process  here  changes,  and  we  find  the  next  figure  of 
the  root  by  dividing  the  absolute  term  .097  by  the  penulti- 
mate co-efficient  --  1.93,  giving  .05  for  the  next  figure  of  the  root. 

We  again  transform  the  equation  into  another  whose  roots 
shall  be  less  by  .05,  and  the  co- efficients  of  the  new  equation 
are   marked   by   three   stars. 

We  then  divide  the  absolute  term,  .010375  by  the  penultimate 
co-efficient,  —  1.5325,  and  obtain  .006,  the  next  figure  of  the 
root :    and   so   on   for   other   figures. 

In  regard  to  the  contractions,  we  may  observe  that,  having 
decided  on  the  number  of  decimal  places  to  which  the  figures 
in  the  root  are  to  be  carried,  we  need  not  take  notice  of 
figures  which  fall  to  the  right  of  that,  number  in  any  of  the 
dividends.  In  the  example  under  consideration,  we  propose  to 
carry  the  operations  to  the  9th  decimal  place  of  the  root ; 
hence,  we  may  reject  all  the  decimal  places  of  the  dividends 
after   the    9th. 

The  fourth  dividend,  marked  by  four  stars,  contains  nine 
decimal   places,  and   the   next   dividend    is   to   contain   no    more. 


CHAP.  XL]  HORNER's  METHOD.  897 

But  the  corresponding  quotient  figure  8,  is  the  fourth  figure 
from  the  decimal  point ;  hence,  at  this  stage  of  the  operation,  all 
the  places  of  the  divisor,  after  the  5th,  may  be  omitted,  since 
the  5th,  multiplied  by  the  4th,  will  give  the  9th  order  of  deci- 
mals. Again :  since  each  new  figure  of  the  root  is  removed 
one  place  to  the  right,  one  additional  figure,  in  each  subsequent 
divisor,  may  be  omitted.  The  contractions,  therefore,  begin  by 
striking  off  the  2  in   the   4th   divisor. 

In  passing  from  the  first  column  to  the  second,  in  the  next 
operation,  we  multiply  by  .0008 ;  but  since  the  product  is  to 
be  limited  to  five  decimal  places,  we  need  take  notice  of  but 
one  decimal  place  in  the  first  column ;  that  is,  in  the  first 
operation  of  contraction,  we  strike  off*,  in  the  first  column,  the 
two  figures  68 ;  and,  generally,  for  each  figure  omitted  in  the 
second  column,  we  omit  two  in  the  first. 

It  should  be  observed,  that  when  places  are  omitted  in  either 
column,  whatever  would  have  been  carried  to  the  last  figure 
retained,  had  no  figures  been  omitted,  is  always  to  be  added 
to  that  figure.  Having  found  the  figure  8  of  the  root,  we  need 
not  annex  it  in  the  first  column,  nor  need  we  annex  any  sub- 
sequent figures  of  the  root,  since  they  would  all  fall  at  the 
right,  among  the  rejected  figures.  Hence,  neither  8,  nor  any 
subsequent  figures  of  the  root,  will  change  the  available  part 
of  the   first   column. 

In  the  next  operation,  we  divide  .000141586  by  1.4772,  omit- 
ting the  figure  8  of  the  divisor :  this  gives  the  figure  9  of  the 
root.  We  then  strike  oflT  the  figures  4.0,  in  the  first  column, 
and  multiplying  by  .00009,  we  form  the  next  divisor  in  the 
second  column,  — -  1.4769,  and  the  next  dividend  in  the  3d 
column,  .000008663.  Striking  off*  5  in  this  divisor,  we  find 
the   next   figure   of  the   root,  which   is   5. 

It  is  now  evident  that  the  products  from  the  first  column, 
will  fall  in  the  second,  among  the  rejected  figures  at  the  right; 
we   need,  therefore,  in  future,  take   no   notice  of  them. 

Omitting  the  right  hand  figure,  the  next  divisor  will  be  1.476, 
and    the   next   figure   of  the   root    8.      Then   omitting   6   in    the 


898 


ELEMENTS   OF  ALGEBRA. 


[CHAP.   XI. 


divisor,  we  obtain  the  quotient  figure  8 :  omitting  7  we  obtain 
6,  and  omitting  4  we  obtain  7,  the  last  figure  to  be  found.  We 
have  thus  found  the  root  x  =  1.356895876  .  .  .  . ;  and  all  similajf 
examples   are  wrought  after  the   same   manner. 

Tlie  next  operation  is  to  find  the  root  whose  initial  figures  ar« 
1.6,  to  nine  decimal  places.  The  operations  are  entirely  similar 
to   those  just   explained. 

We   find  for  the  second  root,  x  =  1.69202141. 

Eor  the  negative  root,  change  the  signs  of  the  second  and 
fourth  terms  (Art.  280),  and  we  have, 


1  -0 

-    7 

-  7  (3.0489173396 

3 

9 
"2 

+  6 
-1 

3 

18 
20 

.814464 

6 

—  .185536 

3 

.3616 

.166382592 

9.0  4 

20.3616 

—  19153408 

4 

.3632 

18791228 

9.0  8 

20.7248 

—362180 

4 

7302 

4 
4 

208875 

9.1  28 

20.79782 

-  153305 

8 

73088 

146212 

9.136 

20.87091 

2 

-7093 

8 

823 

0 

6266 

|..|9.1|44 

20.87914 

2 

-827 

823 

0 

626 

20.8873 

7 
9 

-201 

188 

20.8874 

6 

-13 

9 

12 

2|0.|8|8|7|5 

1 

4.  Find  the  roots  of  the  equation 

x^  f  lla;2~102;r 

+ 

'  1^ 

n  r=0. 

CHAP.  XL]  HORNER'S  METHOD.  399 

The  functions  are 

X  =z    x^  +  11.t2  -  102^  +  181 
Xi  =  3^2  _|.  22a;   -  102 
X,  =  122a;  —  393 

^.=  +  ; 

and 'the  signs  of  the  leading  terms  are   all  -|-  ;    hence,  the  sul> 
sfcitution  cf  —  oc  and  +  ^  must  give  three  real  roots. 

To  discover  the  situation  of  the  roots,  we  make  the  substitu- 

tiODS 

X  z=zO    which  gives    H [-    two  variations 

Xzzzl  "  H \-  " 

a?  =  2  "  H (-  « 

a;  =  3  "  H 1-  " 

a?  =  4  "  +  +  +  +no  variation ; 

hence  the  two  positive  roots  are  between  3  and  4,  and  we  must 
therefore  transform  the  several  functions  into  others,  in  which  z 
shall  be  diminished  by  3.     Thus  we  have  (Art.  314), 
r  =        y3  +  20y2  -  9y  +  1 
Y,  =      3y2  +  40y   -  9 
F,  =  122y   -  27 
Fs=   + 
Make  the  following  substitutions  in  these  functions,  \iz. : 

2/  =  0    signs    H 1-    two  variations 

y  =  .1        "       + + 

y  =  .2      "      + + 

y  =  .3       "       +  +  +  +    no  variation  ; 
hence,  the   two   positive  roots  are  between  3.2  and  3.3,  and  wp 
must  again  transform  the   last  functions   into   others,  in  which  y 
fthall  be  diminished  by  .2.     Effecting  this  transformation,  we  have 
Z  z=:        ^3  +  20.6^2  _  .882  +  .008 
Zi=      Sz^  +  4:l.2z  -.88 
Za  =  1222  -    2.6 
Z,=  +. 


400  •    ELEMENTS   OF   ALGEBRA.  [CHAP.   XL 

Let  z  =  0     then  signs  are    -{ f-     two  variations, 

z  =  .Ol         "         "  + -f 

2  =  .02         "         "  h    one  variation, 

2  =  .03         "         "  +  +  +  +     no  variation  ; 

Hence  we  have  3.21  and  3.22  for  the  positive  roots,  and  the  sum 
of  the  roots  is  —  11  ;  therefore,  —  11  —  3.21  —  3.22  =  —  1X.4, 
IS  the  negative  root,  nearly.  ' 

For   the   positive  root,  whose  initial  figures  are  3.21,  we  have 
X  =  3.21312775 ; 
and  for  the  root  whose  initial  figures  are  3.22,  we  have 

X  =  3.229522121  ; 
and  for   the  negative  root, 

x=  ^  17.44264896. 

EXAMPLES. 

1.  Find  a  root  of  the  equation  x^  +  x^  +  x  —  100  =  0. 

Ans.   4.2644299731. 

2.  Find  the  roots  of  the  equation  ic*  —  12;^^  +  12:r  —  3  =  0. 

'+  2.858083308163 

.     +  .606018306917 
Ans,    'I 

+    .443276939605 
-  3.907378554685. 

3.  Find  the  roots  of  the  equation  x^  —  Sx^  +  14.i;2  -^  4^:  _  8=0 

+  5.2360679775 
+  .7639320225 
+  2.7320508075 
-  .7320508075. 


Ans,    -i 


%    Find  the  roots  of  the  equation 

ifS-102;3  +  6a;+l  =0. 

-  3.0653157912983 

—  .6915762804900 
Ans.    ^    -  .1756747992883 

+  .8795087084144 
,  +  3.0530581626622. 


rt. 


'.rf 


^'^^^  C  .^.■. 


/L 


I 


y^ 


^A^Ti'U't^-^ 


